MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "MATHEMATICS FOR ENGINEERING BASIC ALGEBRA"

Transcription

1 MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals. The tutorial contains the following. On completion of this tutorial you should be able to do the following. Manipulate and solve simple algebraic equations. Solve simultaneous equations by graphical methods. Solve simultaneous equations by the method of elimination. Solve simultaneous equations by the method of matrices. Define the roots of a quadratic equation. Factorise quadratic equations. Solve quadratic equations by graphical methods. Solve quadratic equations by the method of completing the root. Solve quadratic equations by use of the quadratic equation. Solve cubic equations by use of the graphical method. D.J.Dunn 1

2 In tutorial 1 you learned some of the elements of algebra and how to manipulate formulae. In this tutorial we will look at some of the types of equations you will have to deal with in your studies. 1. SIMPLE EQUATIONS A simple equation contains one unknown quantity with powers no higher than 1. It should be easy to solve the unknown by applying the algebraic rules in tutorial 1. The solution of a simple equation is called the root. WORKED EXAMPLE No.1.1 The subscription for membership of an organisation is x Euros. 54 members joint from Ireland, 76 from Germany, 32 from France and 48 from Spain. The total sum received is 420 Euros. Write out an equation for the sum and determine the subscription. Adding the subscriptions must produce 54x + 76x + 32x + 48 x = 420 Simplify by adding all the x terms. 210x = 420 Divide both sides by 210 x = 420/210 = 2 Euros. WORKED EXAMPLE No. 1.2 A restaurant owner buys 24 boxes of tea bags. Each box contains an unknown number of tea bags. He uses 10 boxes on one day and 8 boxes the next day. He then checks and finds he has 150 tea bags left. How many tea bags were there in each box? Let there be n tea bags in each box. The total number of tea bags bought must be 24n The number of boxes used was = 18 The number of tea bags used was 18n The number of tea bags left must be 24n 18n = 150 Evaluate 6n = 150 Divide both sides by 6 and n = 150/6 = 25 There were 25 bags in each box. SELF ASSESSMENT EXERCISE No Solve the unknown in each of the following. 8x - 2x + 3 = 3x +12 5m 5 + 2m = m A cinema sells 25 tickets for screen one, 68 tickets for screen two and 51 tickets for screen three. The price is the same for all tickets and the takings come to 504. What is the price of the tickets? D.J.Dunn 2

3 HARDER EXAMPLES WORKED EXAMPLE No. 1.3 Solve x in the following equation. 5 x x 3 12 Add 7 to both sides 5 4 x x Subtract 5 4 x x x Simplify x 19 Simplify x Simplify x 19 Finally solve x WORKED EXAMPLE No. 1.4 Solve x in the following equation. 2(x 5) - 3(x + 7) = x + 12 Multiply out brackets (2x 10) - (3x + 21) = x + 12 Remove brackets noting signs 2x 10-3x - 21 = x + 12 Gather all x terms on one side and all numbers on the other = -2x + 3x + x -43 = 2x x = -43/2 = -21 ½ WORKED EXAMPLE No. 1.5 A rectangular plate is 200 mm long. A strip 50 mm wide is cut off one end. A second strip 15 mm wide is cut off the other end. The remaining plate weight 2025 g. Find the width of the plate if the plate weighs 0.12 g per square mm. Let the plate be x mm wide. Area = 200x The area cut off is 4.5x and 11.5x Area left is 200x 50x 15x The weight of the remaining plate is (200x 50x 15x)(0.12) Equate to known weight (200x 50x 15x)(0.12) = 2025 Divide both sides by 0.09 (200x 50x 15x) = 2025/0.12 = Add the x terms 135x =16875 x = 125 mm D.J.Dunn 3

4 SELF ASSESSMENT EXERCISE No H H 1. Solve the unknown in the following equation. 12 H Solve the unknown in the following equation. 6(w + 2) 3(w 2) = 2w 3. A train travels from its starting station to another station x km distance along the track at an average speed of 50 km/h. It waits half an hour in the station before making the return trip at an average speed of 80 km/h. The total time taken for journey is 4 hours. What is the distance between the two stations? 4. A metal strip is x mm wide. It is used to make the four sides of a rectangular box measuring 200 mm by 100 mm. The area of the strip mm 2. What is the width of the strip? D.J.Dunn 4

5 2. SIMULTANEOUS EQUATIONS Consider a vehicle that starts moving from the starting post as shown and accelerates at a constant rate of a m/s 2. After travelling a short distance it passes point 0 with velocity u 0 and a stop watch is started. It accelerates a further distance s 1 and passes point 1 with velocity u 1 and the time is noted as t 1. It accelerates to the next timing point and passes it at velocity u 2 and the time is t 2. The distance travelled from the start of timing is s 2. The equation relating these quantities is known to be s = u 0 t + at 2 /2 We can use the equation twice and have: s 1 = u 0 t 1 + at 1 2 /2 and s 2 = u 0 t 2 + at 2 2 /2 This is an example of simultaneous equations where the same equation is used twice with two sets of values to solve two unknown values. The unknown values are u 0 and a. For a given value of time, the distance travelled is a function of u 0 and a so we could say s = f(u 0, a). We will do a solution for this problem later on (worked example 4). In general simultaneous equations take the form Ax + By = f 1 (x,y) Cx + Dy = f 2 (x,y) Where A, B, C and D are known coefficients. This tutorial only deals with two unknown variables. If we had three unknown variables we would need three simultaneous equations and so on. GRAPHICAL We can rearrange our equation to make x the subject and then y the subject as follows. x = (f 1 (x,y) By) A or x = (f 2 (x,y) Dy) C y = (f 1 (x,y) Ax) B or y = (f 2 (x,y) Cx) D We could plot y against x for either pair of equations and determine the point where graphs cross and hence have the same values of x and y. Graphical solutions are laborious and only really suitable for two simultaneous equations but the following example is useful. D.J.Dunn 5

6 WORKED EXAMPLE No. 2.1 Solve x and y given the following simultaneous equation f(x,y) 1 = 39 = x + 7y...(1) f(x,y) 2 = 23 = 2x + 3y..(2) Rearrange to make y the subject (you could make x the subject). y = (39 x)/7.(3) y = (23 2x)/3 (4) Plot x against y using both equations and we get: We find that x = 4 and y = 5 satisfies both equations. SUBSTITUTION METHOD This is a suitable method when there are only two unknown variables but difficult for three or more. We use one equation to find x in terms of y and substitute it into the other equation to solve y or vice-versa. This is best demonstrated by doing the last example again. WORKED EXAMPLE No. 2.2 Solve x and y given the following simultaneous equations: f(x,y) 1 = 39 = x + 7y..(1) f(x,y) 2 = 23 = 2x + 3y..(2) From equation (1) x = 39 7y Substitute this into equation (2) and solve y 23 = 2(39 7y) + 3y 23 = 78 14y + 3y 23 = 78 11y 11y = = 55 y = 55/11 = 5 Now substitute for y into either (1) or (2) to find x. Doing both is a check on the answer. 39 = x + 7y = x + 7(5) = x + 35 x = 4 23 = 2x + 3y = 2x + 3(5) = 2x x = 8 x = 4 D.J.Dunn 6

7 WORKED EXAMPLE No. 2.3 Solve x and y given the following simultaneous equation f(x,y) 1 = 9 = 3x + 5y..(1) f(x,y) 2 = 5 = 2x + 3y..(2) From equation (1) x = (9 5y)/3 Substitute this into equation (2) and solve y 5 = 2(9 5y)/3 + 3y 5 = 6 10y/3 + 3y 5 = 6 y/3 y = 3 Now substitute for y into either (1) or (2) to find x. Doing both is a check on the answer. 9 = 3x + 5y = 3x + 5(3) = 3x x = -6 x = -2 5 = 2x + 3y = 2x + 3(3) = 2x + 9 2x = -4 x = -2 WORKED EXAMPLE No. 2.4 The distance travelled by an object s (metres) is related to the initial velocity u 0 (m/s), the acceleration a (m/s 2 ) and time t (s) by the equation s = u 0 t + at 2 /2 A stop watch is started as the object passes a point such that s = 0 when t = 0. It takes 2 seconds for the object to travel 14 meters from the point and 5 seconds to travel 50 m from the point. Determine the initial velocity and the acceleration. First set up two equations using the information supplied. 14 = 2u 0 + a(2 2 )/2 14 = 2u 0 + 2a..(1) 50 = 5u 0 + a(5 2 )/2 50 = 5u a..(2) From equation (1) u 0 = (14 2a)/2 = 7 - a Substitute this into equation (2) and solve a 50 = 5(7 - a) a = 35-5a a = a 7.5a = = 15 a = 2 m/s 2 Now substitute for y into either (1) or (2) to find u. Doing both is a check on the answer. 14 = 2u 0 + 2a = 2u = u 0 = 5 m/s 50 = 5u a = 5u (2) = 5u u 0 = 25 u 0 =5 m/s D.J.Dunn 7

8 ELIMINATION METHOD Many people find this method easier than substitution. The method requires that we make the coefficient of one of the variables the same by multiplying each term by a suitable number. We then add or subtract the two equations to eliminate one of the variables. A worked example shows this best. WORKED EXAMPLE No. 2.5 A resistance thermometer has a resistance R = 101Ω at a temperature θ = 20 o C and 103Ω at 60 o C. The law relating resistance and temperature is R = Ro + αθ where Ro is the resistance at 0 o C and R is the resistance at any other temperature. α is the temperature coefficient of resistance. Calculate the temperature Ro and α. Ro + 20α = 101..(1) Ro + 60α = 103..(2) Just subtract the equations Ro + 20α = 101 Ro + 60α = 103 Subtract 0-40α = -2 α = -2/-40 = 0.05 Substitute for α in (1) Ro + 20(0.05) = 101 Ro = = 100Ω The relationship is R = θ We might use the same process after making x or y the subject as shown in the next example. WORKED EXAMPLE No. 2.6 Repeat worked example 1 using the elimination method. 39 = x + 7y..(1) 23 = 2x + 3y..(2) Make x the subject in both cases. x = 39 7y (3) x = (23 3y)/2 x = y (4) Subtract 0 = y Solve y 5.5y = 27.5 y = 5 78 = 2x + 14y Substitute for y in any equation and x is 4 as before. D.J.Dunn 8

9 WORKED EXAMPLE No. 2.7 Solve x and y given. 3x + 2y = 12.. (1) x + 3y = (2) Multiply each term in equation (2) by 3 so that the coefficient of x is the same in both. We then have: 3x + 2y = 12 3x + 9y = 33 Subtract 0-7 y = -21 y = -21/-7 = 3 Substitute for y in (1) and 3x + 2(3) = 12. 3x = 6 x = 2 Check in equation (2) 3x + 9y = 33 3(2) + 9(3) = 33 correct WORKED EXAMPLE No. 2.8 What are the values of x and y that satisfies both the following equations. x/7 y/2 = -3..(1) x/3 + y/4 = 10..(2) Multiply each term in (1) by 2 and each term in (2) by 4. We have: 2x/7 y = -6 4x/3 + y = 40 Add 34x/ = 34 note that 2/7 + 4/3 = 34/21 34x/21 = 34 x = 21 Substitute into (1) 21/7 y/2 = -3 3 y/2 = -3 6 = y/2 y = 12 Check in (2) x/3 + y/4 = 10 21/3 + 12/4 = 10 correct D.J.Dunn 9

10 SELF ASSESSMENT EXERCISE No. 2.1 Solve the variables in the following simultaneous equations. 1. 3x + 5y = 6 and 2x + 3y = 3 (Answers x = -3, y = 3) 2. 7x + 4y = 20 and 3x + 3y = 6 (Answers x = 4, y = -2) 3. 5x + 5y = -40 and x + 3y = -48 (Answers x = 12, y = -20) 4. x + y = 3/4 and 2x + 3y = 7/4 (Answers x = ½, y = 1/4) 5. x + 2 y = 1.9 and 2x + 5y = 4 (Answers x = 3/2, y = 1/5) 6. 3x - 5 y = 204 and 4x + 5y = 412 (Answers x = 88, y = 12) 7. x - 5 y = 95 and 2x + y = 245 (Answers x = 120, y = 5) MATRIX METHOD This method should only be used if you have a good understanding of matrix theory. The main use of this method is that it enables the solution of problems with more than two unknown variables and because the method is based on strict rules, it is suitable for use in computer programmes. Here is a basic example that might serve as an introduction to matrix theory. WORKED EXAMPLE No. 2.9 What are the values of x and y that satisfies both the following equations. 3x + 2y = 22..(1) 2x + 3y =23..(2) Create a matrix of the coefficient and column vectors based on the variables and the function values. The object is to solve the column vector X. The simultaneous equations may be represented as A X = b Next we need to find the inverse matrix A -1 For a 2 x 2 matrix the rule is : D.J.Dunn 10

11 Put in the numbers The solution is given by the product X = A -1 b The rule for multiplying is a 4 x 4 matrix by a single column is Putting in the numbers we get: The solution is x = 4 and y = 5 as before. The matrix method looks difficult but matrix theory is very important in modern Engineering methods and should be thoroughly studies for advanced courses. The example is used only as an introduction. The rules are much more complicated for larger arrays. You might care to solve some of the previous examples with the above model. D.J.Dunn 11

12 3. QUADRATIC EQUATIONS Most relationships in Engineering and Science are anything but proportional and many are quadratic. This means that the function contains a highest power of 2. Take a simple case y = f(x) = 64 = x 2 To find the value of x that satisfies the equation we simply take the square root. x = 64 Now there are two possible answers 8 and -8 because a minus number squared is positive. An immediate problem arises when we look at the case y = -64 = x 2 Taking the square root this time gives x = -64 and you probably don t know how to deal with the roots of negative numbers and we wont cover it here. QUADRATIC EQUATIONS and ROOTS Quadratic equations have the general form:- y = f(x) = 0 = ax 2 + bx + c a, b and c are the coefficient of x 2 x 1 and x o respectively. Of course we don t normally bother writing x 1 as x 1 = x and we don t normally write x o as x o =1. If y is not zero, we only need to subtract y from both sides of the equation to produce the required form. There are normally two values of x that satisfy the function and this may be seen by plotting a graph. Consider the case y = f(x) = 2x 2-4x 6 If we plot x against y over a suitable range we get the graph shown. From the graph we can pick off the two values of x that correspond to any value of y but it is of particular interest to find the values of x when y is zero. These are called the ROOTS OF THE EQUATION and in the example they are x = -1 and x = 3. It is always a good idea to check that putting these values into the equation produces y = 0. x = -1 y = 2(-1) 2-4(-1) 6 = = 0 x = 3 y = 2(3) 2-4(3) 6 = = 0 It is nice when the roots are whole numbers (integers) but this is not usually the case. Note that quadratic equations do not always cross the x axis and the solution to these requires advanced studies. D.J.Dunn 12

13 FACTORISATION You should know that an integer will factorise such that the product of the factors give the original number. For example the factors of 8 may be (8)(1) = 8 or (4)(2) = 8. In the case of equations, the factors are not numbers but expressions. A quadratic equation often factorises into two parts such that the sum of the two parts produces the original equation. Knowing how to factorise an equation takes practice. The first step in understanding this is see how multiplying two factors gives the original equation. Suppose that the two factor are (2x + 2) and (x - 3). Let s multiply them. Here s how. The two factors of y = 2x 2-4x 6 are hence (2x + 2) and (x - 3). The trouble is that we need to do the reverse. Given an equation, find the factors. This rule helps us. Even with this rule, you need to play around with the possibilities before you arrive at the correct answer. The problem is easier if the coefficient a = 1. In the above example, we could simply divide by 2 to get x 2-2x 3 and the factors are (x + 1)(x - 3). Note that the -2 is obtained by adding 1 and -3 and this always works. WORKED EXAMPLE No. 3.1 Solve f(x) = 0 = 6x 2 + 8x + 2 Divide by 6 to get 0 = x 2 + (8/6)x + (2/6) Make two brackets (x + A)( x + B) A B = 2/6 and A + B = 8/6 so make A = 1 and B = 2/6 (x + 1)(x + 2/6) = 0 x = -1 or x = -2/6 = -1/3 Check them out in the original equation x = -1/3 f(x) = 6(-1/3) 2 + 8(-1/3) + 2 = 6/9 8/3 + 2 = 2/3 8/3 + 2 = -6/3 + 2 = 0 x = -1 f(x) = 6(-1) 2 + 8(-) + 2 = = 0 It takes practice to do this quickly and it only works if the numbers are kind. D.J.Dunn 13

14 WORKED EXAMPLE No. 3.2 Solve f(x) = 0 = 2x 2 + 3x - 5 Factorise to get two brackets. We know we must have 2x and x (2x + A)( x - B) = 0 We know AB must be -5 so try 1 and -5 (2x + 1)( x - 5) = 0 The middle term must be the sum of x and -10x and this gives -9x which is wrong. Try 5 and -1 (2x + 5)( x - 1) = 0 The middle term must be the sum of 5x and -2x and this gives 3x which is correct. (2x + 5)( x - 1) = 0 so the roots are x = -5/2 and x = 1 Check it in the original equation. x = -5/2 2(-5/2) 2 + 3(-5/2) - 5 = 25/2 15/2 5 = 0 correct x = 1 2(1) 2 + 3(1) - 5 = = 0 correct SELF ASSESSMENT EXERCISE No. 3.1 Solve the following by factorisation. 1. 2x 2-5x 3 = 0 2. x 2 + 2x 8 = 0 3. x 2-2x 3 = x 2-20x 3 = 0 PERFECT SQUARE ROOTS AND COMPLETING THE SQUARE When the two factors are the same, we have a perfect square root. Consider the case x 2 6x + 9 = 0 If we factorise x 2 6x + 9 we get (x 3) (x 3). We could write x 2 6x + 9 = (x 3) 2 = 0 and there is only one solution x = 3, and this is the perfect root. Factorisation is easier if we can change the equation so that it has a perfect square root. This is best demonstrated with an example. WORKED EXAMPLE No. 3.3 Solve x 2 6x + 8 = 0 If the last number was 9 we would have a perfect square root. We can do this by manipulating the equation to: x 2 6x = 0 or x 2 6x + 9 = 1 Factorise the left side into (x - 3)(x - 3) or (x - 3) 2 Now solve by taking the square root. (x - 3) = 1 = ±1 so x = 4 or x = 2 Check x = 4 x 2 6x + 8 = (4) + 8 = = 0 so 4 is the root of the original equation. Check x = (2) + 8 = = 0 so 2 is the root of the original equation. D.J.Dunn 14

15 SELF ASSESSMENT EXERCISE No. 3.2 Find the roots of the following by completing the square. 1. x 2 + 6x + 5 = 0 2. x x + 20 = 0 3. x 2-8x + 11 = 0 5. x 2-4x - 3 = 0 QUADRATIC FORMULAE A method of finding the solution to quadratic equations that is most reliable and widely used is given here without proof. Arrange the equation into the form ax 2 + bx + c = 0 b b 2 4ac The two solutions are then given by x and this is the quadratic formula that 2a should be memorised. Most scientific/engineering calculators can find the solution direct but this formula is useful in advanced studies. WORKED EXAMPLE No. 3.4 Solve 2x 2 4x - 5= 0 a = 2 b = -4 c = -5 b x 2 b 4ac 2a (-4) (-4) 2(2) 4(2)(-5) x x or x 2.871or Check the answers 2(2.871) 2 4(2.871) - 5= 0 2(-0.871) 2 4(-0.871) - 5= 0 2 SELF ASSESSMENT EXERCISE No. 3.3 Find the roots of the following by completing the square. 1. 3x 2 + 5x + 2 = 0 (Answer -1 and ) 2. 5x 2-2x - 4 = 0 (Answer and ) 3. -2x 2-2x + 5 = 0 (Answer and 1.158) 4. -5x 2 + 3x + 7 = 0 (Answer and 1.521) D.J.Dunn 15

16 4. CUBIC EQUATIONS Suppose wish to solve x in the equation 2x x 2-20x 50 = 0 The solution is the roots of f(x) = 2x x 2-20x 50 To solve x we must plot f(x) = y = 2x x 2-20x 50 and see if any value of x produces a result y = 0 The plot is shown. The graph crosses the y axis three times so there are three values of x that give a solution and these are x = -6, x = and 2.59 Check them out to see if they are correct. x = -6 2x x 2-20x 50 = -2 so the answer is not exact but close. x = x x 2-20x 50 = so the answer is not exact but close. x = x x 2-20x 50 = 0.03 so the answer is not exact but close. The graphical method will not usually give exact answers depending on the scale of the graph. Many cubic plots do not cross the y axis and there may be three, two, one or no real answers that satisfy the equation. We can rearrange the equation to make x 3 the subject and plot two graphs like before. In this case we have : x 3 = (- 10x x + 50)/2 = (- 5x x + 25) If we let f 1 (x) = x 3 and f 2 (x) = (- 5x x + 25) and plot both functions against x, the point where the graphs cross is the point where the functions are equal and give the answer for x. D.J.Dunn 16

17 WORKED EXAMPLE No. 4.1 Solve the following equation graphically. (4x 3 3x 2-36x + 27) = 4.67(x+1) Rearrange to make two functions that can be plotted. Let f 1 (x) = (4x 3 3x 2-36x + 27) and f 2 (x) = 4.67(x + 1) Plot both over a suitable range and pick off the answers from the points where the graphs cross (the common abscissa) x = -3.1,x = 0.56 and x = 3.3 WORKED EXAMPLE No. 4.2 Solve the following equation graphically. (x 2 6)(4 + x 2 ) = 32x If we multiply out the brackets we get a quartic equation x 4-6x 2-28x -24 = 0 This example is to show how a graphical method may be used for any equation if the right approach is used. A solution is to rearrange the equation to make two functions that can be plotted. This is the method used here. Rearrange to the form (x 2 6) = 32x/(4 +x 2 ) Let f 1 (x) = (x 2 6) and f 2 (x) = 32x/(4 +x 2 ) Plot both over a suitable range and pick off the answers x = and x = 3.6 D.J.Dunn 17

18 SELF ASSESSMENT EXERCISE No Solve the following equations. x 2 5.2x = 0 (Answers x = 3.2 or 2) 3 x 2-20 log(x) = 0 (Answer x = 2.15) x x x = 0 (Answers x = -2 or 1.2 or 2.67) 2. A closed metal cylindrical canister has a mean radius R and length L. The surface area of the metal used is given by the formula A = 2π(R 2 + RL) Given that the area is 1000 mm 2 and the length is 20 mm what is the radius? (Answers 6.1 mm, the negative answer is ignored as it is not physically possible) 3. A cylindrical vessel of diameter D and length L has hemispherical ends. The volume of the vessel is hence given by V = π(dl + D 3 /6). If the volume is 200 x 10 3 mm 3 and the length L is 50 mm, what is the diameter? (Answer mm, the other answer mm is obviously only theoretical and not physical) 4. Solve the three roots of the equation y = 5x 3 +10x 2-2x 6 (Answer , and 0.735) SAE x = 3 m = 3 1/ SAE S SA H H 4H 3H 6H H 1. H H w w + 6 = 2w w = Time taken to complete the first journey is x/50 Time spent waiting = ½ Time taken for the return journey is x/80 Total time = x/50 + ½ + x/80 = 4 x/50 + x/80 = 3 ½ =7/2 x(1/50 + 1/80) = 7/2 x( )/4000 = 7/2 x (130/4000) = 7/2 x = (28000/260) = km mm D.J.Dunn 18

19 S.A.E (Answers x = -3, y = 3) 2. (Answers x = 4, y = -2) 3. (Answers x = 12, y = -20) 4. (Answers x = ½, y = 1/4) 5. (Answers x = 3/2, y = 1/5) 6. (Answers x = 88, y = 12) 7. (Answers x = 120, y = 5) S.A.E (3 and ½) 2. (2 and 4) 3. (3 and 1) 4. (3 and -1/7) S.A.E (Answer -1 and -5) 2. (Answer -2 and -10) 3. (Answer and 6.236) 6. (Answer and ) S.A.E (Answer -1 and ) 2. (Answer and ) 3. (Answer and 1.158) 4. (Answer and 1.521) S.A.E (Answers x = 3.2 or 2) (Answer x = 2.15) (Answers x = -2 or 1.2 or 2.67) 2. (Answers 6.1 mm, the negative answer is ignored as it is not physically possible) 3. (Answer mm, the other answer mm is obviously only theoretical and not physical) 4. (Answer , and 0.735) D.J.Dunn 19

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

Roots of quadratic equations

Roots of quadratic equations CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients

More information

Key. Introduction. What is a Quadratic Equation? Better Math Numeracy Basics Algebra - Rearranging and Solving Quadratic Equations.

Key. Introduction. What is a Quadratic Equation? Better Math Numeracy Basics Algebra - Rearranging and Solving Quadratic Equations. Key On screen content Narration voice-over Activity Under the Activities heading of the online program Introduction This topic will cover: the definition of a quadratic equation; how to solve a quadratic

More information

3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes

3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

More information

Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

More information

Equations, Inequalities & Partial Fractions

Equations, Inequalities & Partial Fractions Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 1 ALGEBRAIC LAWS This tutorial is useful to anyone studying engineering. It uses the principle of learning by example. On completion of this tutorial

More information

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if 1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

More information

5.4 The Quadratic Formula

5.4 The Quadratic Formula Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

More information

Year 9 set 1 Mathematics notes, to accompany the 9H book.

Year 9 set 1 Mathematics notes, to accompany the 9H book. Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

More information

is identically equal to x 2 +3x +2

is identically equal to x 2 +3x +2 Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3

More information

Teaching & Learning Plans. Quadratic Equations. Junior Certificate Syllabus

Teaching & Learning Plans. Quadratic Equations. Junior Certificate Syllabus Teaching & Learning Plans Quadratic Equations Junior Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve.

More information

MATHEMATICS FOR ENGINEERS BASIC MATRIX THEORY TUTORIAL 2

MATHEMATICS FOR ENGINEERS BASIC MATRIX THEORY TUTORIAL 2 MATHEMATICS FO ENGINEES BASIC MATIX THEOY TUTOIAL This is the second of two tutorials on matrix theory. On completion you should be able to do the following. Explain the general method for solving simultaneous

More information

3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes

3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same

More information

Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

More information

Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

More information

LINEAR EQUATIONS 7YEARS. A guide for teachers - Years 7 8 June The Improving Mathematics Education in Schools (TIMES) Project

LINEAR EQUATIONS 7YEARS. A guide for teachers - Years 7 8 June The Improving Mathematics Education in Schools (TIMES) Project LINEAR EQUATIONS NUMBER AND ALGEBRA Module 26 A guide for teachers - Years 7 8 June 2011 7YEARS 8 Linear Equations (Number and Algebra : Module 26) For teachers of Primary and Secondary Mathematics 510

More information

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2

Unit 1: Polynomials. Expressions: - mathematical sentences with no equal sign. Example: 3x + 2 Pure Math 0 Notes Unit : Polynomials Unit : Polynomials -: Reviewing Polynomials Epressions: - mathematical sentences with no equal sign. Eample: Equations: - mathematical sentences that are equated with

More information

The Not-Formula Book for C1

The Not-Formula Book for C1 Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

More information

REVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012

REVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit

More information

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 4 AREAS AND VOLUMES This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.

More information

1 of 43. Simultaneous Equations

1 of 43. Simultaneous Equations 1 of 43 Simultaneous Equations Simultaneous Equations (Graphs) There is one pair of values that solves both these equations: x + y = 3 y x = 1 We can find the pair of values by drawing the lines x + y

More information

Park Forest Math Team. Meet #5. Algebra. Self-study Packet

Park Forest Math Team. Meet #5. Algebra. Self-study Packet Park Forest Math Team Meet #5 Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number

More information

Module M1.4 Solving equations

Module M1.4 Solving equations F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module M.4 Opening items. Module introduction.2 Fast track questions. Ready to study? 2 Solving linear equations 2. Simple linear equations

More information

Mathematics Chapter 8 and 10 Test Summary 10M2

Mathematics Chapter 8 and 10 Test Summary 10M2 Quadratic expressions and equations Expressions such as x 2 + 3x, a 2 7 and 4t 2 9t + 5 are called quadratic expressions because the highest power of the variable is 2. The word comes from the Latin quadratus

More information

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

Lecture 7 : Inequalities 2.5

Lecture 7 : Inequalities 2.5 3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of

More information

Step 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides:

Step 1: Set the equation equal to zero if the function lacks. Step 2: Subtract the constant term from both sides: In most situations the quadratic equations such as: x 2 + 8x + 5, can be solved (factored) through the quadratic formula if factoring it out seems too hard. However, some of these problems may be solved

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

Chapter 8. Quadratic Equations and Functions

Chapter 8. Quadratic Equations and Functions Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property

More information

The Quadratic Formula

The Quadratic Formula Definition of the Quadratic Formula The Quadratic Formula uses the a, b and c from numbers; they are the "numerical coefficients"., where a, b and c are just The Quadratic Formula is: For ax 2 + bx + c

More information

MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 BASIC INTEGRATION

MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 BASIC INTEGRATION MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 ASIC INTEGRATION This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning

More information

Further Maths Matrix Summary

Further Maths Matrix Summary Further Maths Matrix Summary A matrix is a rectangular array of numbers arranged in rows and columns. The numbers in a matrix are called the elements of the matrix. The order of a matrix is the number

More information

Mth 95 Module 2 Spring 2014

Mth 95 Module 2 Spring 2014 Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

More information

FACTORING QUADRATICS 8.1.1 and 8.1.2

FACTORING QUADRATICS 8.1.1 and 8.1.2 FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

More information

MECHANICS OF SOLIDS - BEAMS TUTORIAL 3 THE DEFLECTION OF BEAMS

MECHANICS OF SOLIDS - BEAMS TUTORIAL 3 THE DEFLECTION OF BEAMS MECHANICS OF SOLIDS - BEAMS TUTORIAL THE DEECTION OF BEAMS This is the third tutorial on the bending of beams. You should judge your progress by completing the self assessment exercises. On completion

More information

2013 MBA Jump Start Program

2013 MBA Jump Start Program 2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

More information

2015 Junior Certificate Higher Level Official Sample Paper 1

2015 Junior Certificate Higher Level Official Sample Paper 1 2015 Junior Certificate Higher Level Official Sample Paper 1 Question 1 (Suggested maximum time: 5 minutes) The sets U, P, Q, and R are shown in the Venn diagram below. (a) Use the Venn diagram to list

More information

Factorising quadratics

Factorising quadratics Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to

More information

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

More information

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts.

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts. Chapter 10-1 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax

More information

x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent.

x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent. Rules of Exponents: If n > 0, m > 0 are positive integers and x, y are any real numbers, then: x m x n = x m+n x m x n = xm n, if m n (x m ) n = x mn (xy) n = x n y n ( x y ) n = xn y n 1 Can we make sense

More information

Algebra I. In this technological age, mathematics is more important than ever. When students

Algebra I. In this technological age, mathematics is more important than ever. When students In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

More information

QUADRATIC EQUATIONS. 4.1 Introduction

QUADRATIC EQUATIONS. 4.1 Introduction 70 MATHEMATICS QUADRATIC EQUATIONS 4 4. Introduction In Chapter, you have studied different types of polynomials. One type was the quadratic polynomial of the form ax + bx + c, a 0. When we equate this

More information

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

More information

7 Quadratic Expressions

7 Quadratic Expressions 7 Quadratic Expressions A quadratic expression (Latin quadratus squared ) is an expression involving a squared term, e.g., x + 1, or a product term, e.g., xy x + 1. (A linear expression such as x + 1 is

More information

9 Matrices, determinants, inverse matrix, Cramer s Rule

9 Matrices, determinants, inverse matrix, Cramer s Rule AAC - Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:

More information

MATHEMATICAL NOTES #1 - DEMAND AND SUPPLY CURVES -

MATHEMATICAL NOTES #1 - DEMAND AND SUPPLY CURVES - MATHEMATICAL NOTES #1 - DEMAND AND SUPPLY CURVES - Linear Equations & Graphs Remember that we defined demand as the quantity of a good consumers are willing and able to buy at a particular price. Notice

More information

QUADRATIC EQUATIONS YEARS. A guide for teachers - Years 9 10 June The Improving Mathematics Education in Schools (TIMES) Project

QUADRATIC EQUATIONS YEARS. A guide for teachers - Years 9 10 June The Improving Mathematics Education in Schools (TIMES) Project 9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project QUADRATIC EQUATIONS NUMBER AND ALGEBRA Module 34 A guide for teachers - Years 9 10 June 2011 Quadratic Equations (Number and Algebra

More information

Algebra Tiles Activity 1: Adding Integers

Algebra Tiles Activity 1: Adding Integers Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting

More information

Section 2-5 Quadratic Equations and Inequalities

Section 2-5 Quadratic Equations and Inequalities -5 Quadratic Equations and Inequalities 5 a bi 6. (a bi)(c di) 6. c di 63. Show that i k, k a natural number. 6. Show that i k i, k a natural number. 65. Show that i and i are square roots of 3 i. 66.

More information

Placement Test Review Materials for

Placement Test Review Materials for Placement Test Review Materials for 1 To The Student This workbook will provide a review of some of the skills tested on the COMPASS placement test. Skills covered in this workbook will be used on the

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

More information

a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2 Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

Level 2 Certificate Further MATHEMATICS

Level 2 Certificate Further MATHEMATICS Level 2 Certificate Further MATHEMATICS 83601 Paper 1 non-calculator Report on the Examination Specification 8360 June 2013 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright

More information

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

MATHS WORKSHOPS. Functions. Business School

MATHS WORKSHOPS. Functions. Business School MATHS WORKSHOPS Functions Business School Outline Overview of Functions Quadratic Functions Exponential and Logarithmic Functions Summary and Conclusion Outline Overview of Functions Quadratic Functions

More information

MATH Fundamental Mathematics II.

MATH Fundamental Mathematics II. MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/fun-math-2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics

More information

CONNECT: Algebra. 3x = 20 5 REARRANGING FORMULAE

CONNECT: Algebra. 3x = 20 5 REARRANGING FORMULAE CONNECT: Algebra REARRANGING FORMULAE Before you read this resource, you need to be familiar with how to solve equations. If you are not sure of the techniques involved in that topic, please refer to CONNECT:

More information

NSM100 Introduction to Algebra Chapter 5 Notes Factoring

NSM100 Introduction to Algebra Chapter 5 Notes Factoring Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

More information

MATH 10034 Fundamental Mathematics IV

MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

More information

Algebra: Real World Applications and Problems

Algebra: Real World Applications and Problems Algebra: Real World Applications and Problems Algebra is boring. Right? Hopefully not. Algebra has no applications in the real world. Wrong. Absolutely wrong. I hope to show this in the following document.

More information

5-6 The Remainder and Factor Theorems

5-6 The Remainder and Factor Theorems Use synthetic substitution to find f (4) and f ( 2) for each function. 1. f (x) = 2x 3 5x 2 x + 14 Divide the function by x 4. The remainder is 58. Therefore, f (4) = 58. Divide the function by x + 2.

More information

Factoring Quadratic Expressions

Factoring Quadratic Expressions Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

More information

Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.

More information

Pre-Calculus III Linear Functions and Quadratic Functions

Pre-Calculus III Linear Functions and Quadratic Functions Linear Functions.. 1 Finding Slope...1 Slope Intercept 1 Point Slope Form.1 Parallel Lines.. Line Parallel to a Given Line.. Perpendicular Lines. Line Perpendicular to a Given Line 3 Quadratic Equations.3

More information

Algebra. Indiana Standards 1 ST 6 WEEKS

Algebra. Indiana Standards 1 ST 6 WEEKS Chapter 1 Lessons Indiana Standards - 1-1 Variables and Expressions - 1-2 Order of Operations and Evaluating Expressions - 1-3 Real Numbers and the Number Line - 1-4 Properties of Real Numbers - 1-5 Adding

More information

is identically equal to x 2 +3x +2

is identically equal to x 2 +3x +2 Partial fractions.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as + for any

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

( % . This matrix consists of $ 4 5  5' the coefficients of the variables as they appear in the original system. The augmented 3  2 2 # 2  3 4& Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

More information

Foundation. Scheme of Work. Year 10 September 2016-July 2017

Foundation. Scheme of Work. Year 10 September 2016-July 2017 Foundation Scheme of Work Year 10 September 016-July 017 Foundation Tier Students will be assessed by completing two tests (topic) each Half Term. PERCENTAGES Use percentages in real-life situations VAT

More information

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

More information

Teaching Quadratic Functions

Teaching Quadratic Functions Teaching Quadratic Functions Contents Overview Lesson Plans Worksheets A1 to A12: Classroom activities and exercises S1 to S11: Homework worksheets Teaching Quadratic Functions: National Curriculum Content

More information

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

Introduction to Finite Systems: Z 6 and Z 7

Introduction to Finite Systems: Z 6 and Z 7 Introduction to : Z 6 and Z 7 The main objective of this discussion is to learn more about solving linear and quadratic equations. The reader is no doubt familiar with techniques for solving these equations

More information

MATHEMATICS SUBJECT 4008/4028

MATHEMATICS SUBJECT 4008/4028 MATHEMATICS PAPER 2 SUBJECT 4008/4028 GENERAL COMMENTS General performance of candidates was poor with the majority in the 0-10 marks group. This shows that candidates were not prepared thoroughly for

More information

P.E.R.T. Math Study Guide

P.E.R.T. Math Study Guide A guide to help you prepare for the Math subtest of Florida s Postsecondary Education Readiness Test or P.E.R.T. P.E.R.T. Math Study Guide www.perttest.com PERT - A Math Study Guide 1. Linear Equations

More information

SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

More information

0.7 Quadratic Equations

0.7 Quadratic Equations 0.7 Quadratic Equations 8 0.7 Quadratic Equations In Section 0..1, we reviewed how to solve basic non-linear equations by factoring. The astute reader should have noticed that all of the equations in that

More information

Order of Operations. 2 1 r + 1 s. average speed = where r is the average speed from A to B and s is the average speed from B to A.

Order of Operations. 2 1 r + 1 s. average speed = where r is the average speed from A to B and s is the average speed from B to A. Order of Operations Section 1: Introduction You know from previous courses that if two quantities are added, it does not make a difference which quantity is added to which. For example, 5 + 6 = 6 + 5.

More information

Review Session #5 Quadratics

Review Session #5 Quadratics Review Session #5 Quadratics Discriminant How can you determine the number and nature of the roots without solving the quadratic equation? 1. Prepare the quadratic equation for solving in other words,

More information

not to be republished NCERT QUADRATIC EQUATIONS CHAPTER 4 (A) Main Concepts and Results

not to be republished NCERT QUADRATIC EQUATIONS CHAPTER 4 (A) Main Concepts and Results QUADRATIC EQUATIONS (A) Main Concepts and Results Quadratic equation : A quadratic equation in the variable x is of the form ax + bx + c = 0, where a, b, c are real numbers and a 0. Roots of a quadratic

More information

Core Maths C1. Revision Notes

Core Maths C1. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

More information

FROM THE SPECIFIC TO THE GENERAL

FROM THE SPECIFIC TO THE GENERAL CONNECT: Algebra FROM THE SPECIFIC TO THE GENERAL How do you react when you see the word Algebra? Many people find the concept of Algebra difficult, so if you are one of them, please relax, as you have

More information

Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula

Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula An equation is a mathematical statement that two mathematical expressions are equal For example the statement 1 + 2 = 3 is read as

More information

Session 3 Solving Linear and Quadratic Equations and Absolute Value Equations

Session 3 Solving Linear and Quadratic Equations and Absolute Value Equations Session 3 Solving Linear and Quadratic Equations and Absolute Value Equations 1 Solving Equations An equation is a statement expressing the equality of two mathematical expressions. It may have numeric

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

1.01 b) Operate with polynomials.

1.01 b) Operate with polynomials. 1.01 Write equivalent forms of algebraic expressions to solve problems. a) Apply the laws of exponents. There are a few rules that simplify our dealings with exponents. Given the same base, there are ways

More information

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

More information

Time Topic What students should know Mathswatch links for revision

Time Topic What students should know Mathswatch links for revision Time Topic What students should know Mathswatch links for revision 1.1 Pythagoras' theorem 1 Understand Pythagoras theorem. Calculate the length of the hypotenuse in a right-angled triangle. Solve problems

More information