QUADRATIC EQUATIONS Use with Section 1.4


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1 QUADRATIC EQUATIONS Use with Section 1.4 OBJECTIVES: Solve Quadratic Equations by Factoring Solve Quadratic Equations Using the Zero Product Property Solve Quadratic Equations Using the Quadratic Formula The Discriminant and its Meaning Solve a Formula for a Specified Variable Check Solutions on the Graphing Calculator Equivalent equations have the same solutions; replace an equation with an equivalent equation by Simplifying expressions Performing the same operation on both sides of the equation Add same number to both sides Subtract same number from both sides Multiply both sides by same nonzero number Divide both sides by same nonzero number Interchange the sides of the equation Factoring one side IF other side is 0 Some other operations (such as squaring both sides) may result in extraneous solutions Quadratic equations Standard form is ax + bx + c = 0 Degree ; expect to find solutions for most quadratic equations! Solve by Factoring Must set equation equal to 0 first!! Recall Zero Product Property: ab = 0 a = 0 or b = 0 (or both) Solve x 5x = 3 algebraically and check graphically* x 5x = 3 x 5x 3 = 0 (x + 1)(x 3) = 0 x + 1= 0 or x 3 = 0 1 x = or x = 3 See calculator corner for check. Square root property : Use with caution! If x = k, then x = k or x = k x = 49 x =± 7 (You can verify that there are two solutions by using the previous method!) (x + ) = 1 x+ = 1 or x+ = 1 x = 1 or x = 3 x 11= 0 x = 11 x =± 11 Quadratic Equations  page 1
2 b ± b 4ac Quadratic formula: x = a ; (set equation = 0) You must know this formula!! The quantity b 4ac is called the discriminant; it predicts the number of real solutions If b 4ac > 0, then the equation has two unequal real number solutions If b 4ac < 0, then the equation has no real number solutions If b 4ac = 0, then there is a repeated real solution (of multiplicity ) Solve x + 4x = algebraically and check graphically* x + 4x + = 0 a = 1; b = 4; c = b 4ac = 16 4(1)() = 8 there are two real and unequal solutions 4± 4 4(1)() 4± ± 8 4± ( ± ) x = = = = = = ± (1) Solutions must be exact! Solve 3x x + 1= 0 algebraically and check graphically* b 4ac = ( ) 4(3)(1) = 4 1 = 8 < 0 ± 4 1 ± 8 x = = 6 6 There is no real solution. Solve 4x 1x + 9 = 0 algebraically and check graphically* b 4ac there are no real solutions = ( 1) 4(4)(9) = 0 there is a single repeated solution 1 ± ( 1) 4(4)(9) 1 ± x = = = = (4) (4) 8 kmv Solve F = r for v Fr = kmv Fr = v km v =± Fr FrkM or ± km km Quadratic Equations  page
3 *CALCULATOR CORNER (TI83 Plus) Use with Section 1.4 One method of checking your answer to an equation is to substitute your answer for each occurrence of x in the original equation: We solved 8x  (x + 1) = 3x  13 in a previous lesson and got x =  4 as the answer. Checking by hand 8x  (x + 1) = 3x x (x + 1) = 3x 13 8( 4) [( 4) + 1] = 3( 4) 13 8( 4) ( 7) = = 5 5 = 5 proves our answer is correct! Checking on the calculator Start by storing 4 as the value of x; your calculator will use this value for x until you tell it to use some other value or you perform some calculation which automatically changes the value of x. We ll talk more about this later.! 4 STO X ENTER Evaluate each side of original equation and see if you get the same result. Isn t that easy Since checking is so simple, you should check your answer to every equation. It is not a solution if it doesn t check! Both of the above methods assume you have an answer to check! The intersection of graphs method on the calculator will serve as a check if you already have an answer, but it also solves the equation for you (always read the directions to see if you are allowed to use only a graphical solution!) Checking graphically on the calculator Caution: It may not be possible to get an exact answer with this method! Go to the Y= screen and enter the sides of the original equation as Y1 and Y A common starting screen size is 6:ZStandard on the ZOOM key. Press 6:ZStandard and ENTER to draw the graph. We are interested in the point of intersection of the two graphs, so we need to change our window settings to show smaller yvalues. The WINDOW key displays the current settings. The window settings may be summarized using the notation [10, 10] x [10, 10]. (These are the intervals for x and y.) Use the arrow key to go down to Ymin and enter! 3 0. Press ENTER to redraw the graph. Now that we can see the intersection, it will be (relatively) easy to find its coordinates. Quadratic Equations  page 3
4 Enter nd CALC and choose 5:intersect, then press ENTER. The graph will be displayed, along with Y1 (the left side of our equation). The calculator is asking you to verify that it has chosen the correct graph. Press ENTER to accept it. The graph will be displayed, along with Y (the right side of our equation). Again the calculator is asking you to verify that it has chosen the correct graph. Press ENTER to accept. Since two graphs may intersect in more than one point, you have to tell the calculator which point of intersection you are seeking. If there is only one point of intersection, this is not a crucial step. Just pressing ENTER right now will work fine, because the calculator will go to the nearest point of intersection. In general, you either use the left or right arrow key to move the cursor near the desired point of intersection, or you type a reasonable guess for x using the key pad. Press ENTER. Your display must say Intersection. If it doesn t, you have not pressed ENTER enough times and you have not found the actual intersection. The display below shows that the solution to our equation is x = 4. The yvalue is not a part of the solution since the original equation has the single variable x in it. You will recognize the yvalue as the check number which appeared in our check above, however. Checks for other equations solved in this section 1 x(1 + x) = (x  1)(x  ) x = 3 Graphed on [10, 10] x [10, 10]. We need to know if the approximation displayed here is actually the fractional solution we got algebraically. Fortunately, your calculator stores the xvalue and yvalue it has found using Intersection. In the home screen you can easily display the value of x by pressing X ENTER. To display the value of y, you must use ALPHA Y ENTER. It is usually possible to convert these decimal approximations to the actual fractions by using Frac on the MATH key. (Since your calculator can only store a finite number of decimal places, a fractional conversion is not always possible.) Quadratic Equations  page 4
5 1 x 5x = 3 x = or x = 3 Graphed on [10, 10] x [10, 10]. You must go through the intersection process twice to get both answers. x = 49 x =± 7 Graphed on [10, 10] x [10, 60, 5]. (Yscale changed to 5) (x + ) = 1 x = 1, 3 Graphed on [10, 10] x [10, 10]. x 11= 0 x = ± 11 Note: Y = 0 is the xaxis! It must be graphed in order to be interactive. Graphed on [10, 10] x [15, 10]. The only way to verify our solution here is to compare the value of x with 11. Since 11 is an irrational number, the calculator cannot change it to a fraction. x + 4x = x = ± Graphed on [10, 10] x [15, 10]. (I forgot to change the settings!) Quadratic Equations  page 5
6 3x x + 1= 0 no real solution Graphed on [5, 5] x [5, 5] with the axes turned off (under FORMAT ) There is no intersection, so there is no real solution. 3 4x 1x + 9 = 0 x = Graphed on [10, 10] x [10, 10]. This intersection is a single point and the calculator sometimes fails to find an intersection in this type of problem. Since the number of pixels on the calculator screen is so limited, the calculator may be unable to get an exact answer. In this display, the is a roundoff error. Because the calculator has inherent deficiencies, it is important to know how to solve problems algebraically whenever possible. So what to do here You should be skeptical of a display such as since it is very close to 1.5. Try substituting x = 1.5 in the original problem. Since it works perfectly, x = 1.5 is the solution! Quadratic Equations  page 6
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