The Quadratic Function


 Lesley Eaton
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1 0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral in an algebraic term e.g. in a 3 the a is the coefficient Discriminant: Part of the quadratic formula, the algebraic epression b  4ac is called the discriminant as its value determines the number and nature of the roots of a quadratic equation Equations reducible to quadratics: Equations that can be reduced to the form: a + b + c = 0 Indefinite: A quadratic function where f ( ) can be both positive and negative for varing values of Maimum value: The maimum or greatest value of a graph for a given domain Minimum value: The minimum or smallest value of a graph for a given domain Negative definite: A quadratic function where f() is alwas negative for all values of Positive definite: A quadratic function where f() is alwas positive for all values of Root of an equation: The solution of an equation
2 Chapter 0 The Quadratic Function 543 INTRODUCTION THE SOLUTION OF QUADRATIC equations is important in man fields, such as engineering, architecture and astronom. In this chapter ou will stud quadratic equations in detail, and look at the relationship between quadratic equations and the graphs of quadratic functions (the parabola). You will stud the ais of smmetr and maimum and minimum values of the quadratic function. You will also look at the quadratic formula in detail, and at the relationships between the roots (solutions) of quadratic equations, the formula and the quadratic function. DID YOU KNOW? Thousands of cla tablets from ancient Bablonia have been discovered b archaeologists. These tablets are from as far back as 000 BC. The show that the Bablonians had mastered man mathematical skills. Geometr, including Pthagoras theorem, was well developed, and geometric problems were often worked out b using algebra. Quadratic equations were used in solving geometr problems. The word quadratic comes from the Latin quadratum, meaning foursided figure. Completing the square and the quadratic formula were both used to solve quadratic equations. The Bablonians also had some interesting approimations for square roots. For eample, = 7. An approimation for that is ver accurate was found on a tablet dating back to 600 BC: = = Graph of a Quadratic Function Ais of smmetr EXAMPLE (a) Sketch the parabola =  4 on the number plane. (b) Find the equation of the ais of smmetr of the parabola. (c) Find the minimum value of the parabola. (a) For the intercept, = 0 i.e. = 04( 0) = 0 For the intercept, = 0 i.e. 0 =  4 = (  4) ` = 0 or  4 = 0 = 4 The ais of smmetr lies halfwa between = 0 and = 4. CONTINUED
3 544 Maths In Focus Mathematics Etension Preliminar Course (b) The ais of smmetr has equation =. (c) Since the parabola is smmetrical about the line =, the minimum value is on this line. Substitute = into the equation of the parabola i.e. =  4( ) =4 So the minimum value is  4. Class Investigation. How would ou find the ais of smmetr for a graph with no intercepts?. How would ou find the ais of smmetr of a graph where the intercepts are irrational numbers? The ais of smmetr of the quadratic function = a + b + c has the equation b = a
4 Chapter 0 The Quadratic Function 545 Proof The ais of smmetr lies midwa between the intercepts. For the intercepts, = 0 i.e. a + b + c = 0 b! b 4ac =   a The coordinate of the ais of smmetr is the average of the intercepts. i.e. b  b  4ac b b 4ac a a = b a = b =  4a b = a The parabola has a minimum value if a 0. The shape of the parabola is concave upwards. Minimum value The parabola has a maimum value if a 0. The shape of the parabola is concave downwards. Maimum value
5 546 Maths In Focus Mathematics Etension Preliminar Course The minimum or maimum value is f b c m a EXAMPLES. Find the equation of the ais of smmetr and the minimum value of the quadratic function = The equation of the ais of smmetr is given b i.e. ` b = a ( 5) =  ( ) 5 = Equation is = a 0 gives a minimum value. 5 5 Minimum value: = c m  5c m+ 5 5 = =5 4 So minimum value is Find the equation of the ais of smmetr and the maimum value of the quadratic function = a 0 gives a maimum value. The equation of the ais of smmetr is given b b = a i.e. = (  3) = 6 ` Equation is = 6 Maimum value: = 3c m + c m = =4 So maimum value is  4.
6 Chapter 0 The Quadratic Function 547 Class Investigation Eamine the graph of = from the above eample. Are there an solutions for the quadratic equation = 0? The minimum or maimum point of the parabola is called the verte. EXAMPLE (a) Find the equation of the ais of smmetr and the coordinates of the verte of the parabola = (b) Find the intercept and sketch the graph. (a) Ais of smmetr: b = a =  # = 3 When = 3 = 3 ] g  ] 3g+ 7 = So the verte is ( 3,  ). (b) For intercept, = 0 = 0 ] g  0 ] g+ 7 = 7  (3, ) The verte is the minimum point of the parabola since a Eercises. B finding the intercepts on the aes, sketch the parabola = +. Find the equation of its ais of smmetr, and the minimum value.. Find the equation of the ais of smmetr and the minimum value of the parabola = Find the equation of the ais of smmetr and the minimum value of the parabola = Find the equation of the ais of smmetr and the minimum value of the parabola =  4.
7 548 Maths In Focus Mathematics Etension Preliminar Course 5. Find the equation of the ais of smmetr and the minimum point of the parabola = Find the equation of the ais of smmetr and the maimum value of the parabola = Find the equation of the ais of smmetr and the maimum point of the parabola = Find the minimum value of = How man solutions does the equation = 0 have? 9. Find the minimum value of = How man solutions does the equation = 0 have? 0. Find the minimum value of = How man solutions does the equation = 0 have?. Find the equation of the ais of smmetr and the coordinates of the verte for each parabola. (a) = (b) = (c) = + 5 (d) = (e) = Find (i) the equation of the ais of smmetr (ii) the minimum or maimum value and (iii) the verte of the parabola. (a) = +  (b) = Find the maimum or minimum point for each function. (a) = + + (b) = (c) f] g = (d) =  (e) f] g = (f) f] g = (g) = (h) = (i) f] g = (j) f] g = For each quadratic function (i) find an intercepts using the quadratic formula. (ii) state whether the function has a maimum or minimum value and find this value. (iii) sketch the function on a number plane. (a) f] g = (b) f] g = (c) = (d) f] g = + (e) =  8 (f) = (g) f] g = (h) f] g = (i) = (j) = (a) Find the minimum value of the parabola = (b) How man solutions does the quadratic equation = 0 have? (c) Sketch the parabola. 6. (a) How man intercepts has the quadratic function f] g = ? (b) Find the minimum point of the function. (c) Sketch the function. 7. (a) Find the maimum value of the quadratic function f] g = (b) How man solutions has the quadratic equation = 0? (c) Sketch the graph of the quadratic function.
8 Chapter 0 The Quadratic Function (a) Sketch the parabola = (b) From the graph, find values of for which (c) Find the domain over which # Sketch = and hence show that for all. 0. B sketching f] g = + +, show that for all.. Show b a sketch that for all.. Sketch = and show that for all. Investigation Could ou tell without sketching the function = if for all? How could ou do this? How could ou know that for all without sketching the graph of f] g = + 7? You will look at this later on in the chapter. Quadratic Inequalities You looked at solving quadratic inequations in Chapter 3 using the number line. You can also solve them using the graph of a parabola. For an curve on a number plane = 0 is on the ais (all values of are zero on the ais) 0 is above the ais (all positive values of lie above the ais) 0 is below the ais (all negative values of lie below the ais) Substituting a + b + c for in the general parabola = a + b + c gives the following results: For the parabola = a + b + c a + b + c = 0 on the ais a + b + c 0 above the ais a + b + c 0 below the ais
9 550 Maths In Focus Mathematics Etension Preliminar Course a 0 a + b + c 0 a + b + c = 0 a + b + c 0 a 0 a + b + c 0 a + b + c = 0 a + b + c 0 EXAMPLES. Solve $ 0. First sketch = showing intercepts ( a 0 so it is concave upwards). For intercepts, = 0 0 = = ]  g]  g  = 0,  = 0 =, =
10 Chapter 0 The Quadratic Function 55 $ 0 on and above the ais So $ 0 on and above the ais. ` #, $. Solve 40. First sketch = 4  showing intercepts ( a 0 so it is concave downwards). For intercepts, = 0 0 = 4  = ] 4  g = 0, 4  = 0 = 0, 4 = above the ais So 40 above the ais. ` 0 4. CONTINUED
11 55 Maths In Focus Mathematics Etension Preliminar Course 3. Solve First sketch =  5 showing intercepts ( a 0 so it is concave upwards). For intercepts, = 0 0 =  5 = ] + 5g]  5g + 5 = 0,  5 = 0 =5, = below the ais So below the ais. ` Further inequations You learned how to solve inequations involving the pronumeral in the denominator b using the number line in Chapter 3. Here we use quadratic inequalities to solve them.
12 Chapter 0 The Quadratic Function 553 EXAMPLES. Solve $. +!  We don t know whether + is positive or negative, but ] + g is alwas positive. We can multipl both sides of the inequation b ] g without changing the inequalit sign. $ + #] + g $ #] + g + + $ ] + g 0 $ ] + g ] + g $ ] + g6] + $ ] + g] +  g $ ] + g] + g + Factorise b taking out + as a common factor. We solve this b sketching the parabola = ] + g] + g. For intercepts: = 0 0 = ] + g] + g + = 0, + = 0 =, = = $ ] + g] + g on and below the ais. However,!  The solution is  # Solve 5. ! We multipl both sides of the inequation b ]  g. CONTINUED
13 554 Maths In Focus Mathematics Etension Preliminar Course Factorise b taking out  as a common factor # ]  g 5# ]  g  4 ]  g 5]  g 0 5]  g  4 ]  g ]  g65]  g ]  g] g ]  g] 0g We solve this b sketching the parabola = ]  g] 0g. For intercepts: = 0 0 = ]  g]  0g  = 0,  0 = 0 =, = ]  g] 0g above the ais. The solution is, Eercises Solve n + n# 0 3. a  a$ # 0 6. t  t p + 4p + 3$ 0 9. m  6m # 0
14 Chapter 0 The Quadratic Function 555. h  7h # k  k $ 0 4. q  9q $ 4.  $ ] + g $ n  n # # + $ t $4t ]  3g] + g $ # 3  $ 4 + Solve the inequations in Chapter 3 using these methods for etra practice.. 3 The Discriminant The values of that satisf a quadratic equation are called the roots of the equation. The roots of a + b + c = 0 are the intercepts of the graph = a + b + c. If = a + b + c has intercepts, then the quadratic equation a + b + c = 0 has real roots. a 0 a 0 Since the graph can be both positive and negative, it is called an indefinite function.
15 556 Maths In Focus Mathematics Etension Preliminar Course. If = a + b + c has intercept, then the quadratic equation a + b + c = 0 has real root a 0 a 0 3. If = a + b + c has no intercepts, then the quadratic equation a + b + c = 0 has no real roots a 0 a 0 Since this graph is alwas positive, it is called a positive definite function. Since this graph is alwas negative, it is called a negative definite function. This information can be found without sketching the graph. Investigation.. Solve the following quadratic equations using the quadratic formula (a) = 0 (b) = 0 (c) = 0 (d) = 0 Without solving a quadratic equation, can ou predict how man roots it has b looking at the quadratic formula?
16 Chapter 0 The Quadratic Function 557 b! b 4ac In the quadratic formula =  , the epression b  4ac is called a the discriminant. It gives us information about the roots of the quadratic equation a + b + = 0. EXAMPLES Use the quadratic formula to find how man real roots each quadratic equation has = 0 b! b 4ac =   a 5! 5 4# # 3 = # 5! 5 =  + 5! 37 =  There are real roots: =  +, = 0 b! b 4ac =   a ( )! ( ) 4# # 4 = #! 5 = There are no real roots since  5 has no real value = 0 b! b 4ac =   a ( )! ( ) 4# # = #! 0 = CONTINUED
17 558 Maths In Focus Mathematics Etension Preliminar Course There are real roots: 0 0 = +,  =, However, these are equal roots. Tis the Greek letter 'delta'. Notice that when there are real roots, the discriminant b  4ac 0. When there are equal roots (or just real root), b  4ac = 0. When there are no real roots, b  4ac 0. We often use D = b  4ac.. If T 0, then the quadratic equation a + b + c = 0 has real unequal (different) roots. a 0 a 0 If T is a perfect square, the roots are rational. If T is not a perfect square, the roots are irrational.. If T = 0, then the quadratic equation a + b + c = 0 has real root or equal roots. a 0 a 0
18 Chapter 0 The Quadratic Function If T 0, then the quadratic equation a + b + c = 0 has no real roots. a 0 a 0 If T 0 and a 0, it is positive If T 0 and a 0, it is definite and a + b + c 0 negative definite and for all. a + b + c 0 for all. We can eamine the roots of the quadratic equation b using the discriminant rather than the whole quadratic formula. EXAMPLES. Show that the equation = 0 has no real roots. T = b  4ac =  4] g] 4g =  3 =3 0 So the equation has no real roots.. Find the values of k for which the quadratic equation k = 0 has real roots. For real unequal roots, T 0. For real equal roots, T = 0. So for real roots, T $ 0. CONTINUED
19 560 Maths In Focus Mathematics Etension Preliminar Course T $ 0 b  4ac $ 0 ]g  4] 5g] kg $ 0 40k $ 0 4 $ 0k 5 $ k 3. Show that for all. If a 0 and T 0, then a + b + c 0 for all. a 0 a = 0 T = b  4ac = ]g  4] g] 4g = 46 = 0 Since a 0 and T 0, for all. 4. Show that the line = 0 is a tangent to the parabola =. For the line to be a tangent, it must intersect with the curve in onl point.
20 Chapter 0 The Quadratic Function It is too hard to tell from the graph if the line is a tangent, so we solve simultaneous equations to find an points of intersection. = ] g = 0 ] g Substitute () into (): = = 0 We don t need to find the roots of the equation as the question onl asks how man roots there are. We find the discriminant. D = b  4ac = 44] g] 4g = 66 = 0 ` the equation has real root (equal roots) so there is onl one point of intersection. So the line is a tangent to the parabola. 0.3 Eercises. Find the discriminant of each quadratic equation. (a) = 0 (b) = 0 (c) = 0 (d) = 0 (e) = 0 (f) + 4 = 0 (g)  + = 0 (h) = 0 (i) = 0 (j) = 0. Find the discriminant and state whether the roots of the quadratic equation are real or imaginar (not real), and if the are real, whether the are equal or unequal, rational or irrational.
21 56 Maths In Focus Mathematics Etension Preliminar Course (a) = 0 (b) = 0 (c) = 0 (d) = 0 (e) = 0 (f) = 0 (g) = 0 (h) = 0 (i)  + = 0 (j) = 0 3. Find the value of p for which the quadratic equation + + p = 0 has equal roots. 4. Find an values of k for which the quadratic equation + k + = 0 has equal roots. 5. Find all the values of b for which + + b + = 0 has real roots. 6. Evaluate p if p = 0 has no real roots. 7. Find all values of k for which ] k + g = 0 has real unequal roots. 8. Prove that for all real. 9. Find the values of k for which + ] k + g + 4 = 0 has real roots. 0. Find values of k for which the epression k + 3k + 9 is positive definite.. Find the values of m for which the quadratic equation  m + 9 = 0 has real and different roots.. If  k + = 0 has real roots, evaluate k. 3. Find eact values of p if p  + 3p = 0 is negative definite. 4. Evaluate b if ] b  g  b + 5b = 0 has real roots. 5. Find values of p for which the quadratic equation + p + p + 3 = 0 has real roots. 6. Show that the line = + 6 cuts the parabola = + 3 in points. 7. Show that the line = 0 cuts the parabola = in points. 8. Show that the line =  4 does not touch the parabola =. 9. Show that the line = 5  is a tangent to the parabola = The line = 3  p + is a tangent to the parabola =. Evaluate p.. Which of these lines is a tangent to the circle + = 4? (a) = 0 (b) = 0 (c) = 0 (d) = 0 (e) = 0 Quadratic Identities When ou use the quadratic formula to solve an equation, ou compare a quadratic, sa, = 0 with the general quadratic a + b + c = 0.
22 Chapter 0 The Quadratic Function 563 You are assuming when ou do this that and a + b + c are equivalent epressions. We can state this as a general rule: If two quadratic epressions are equivalent to each other then the corresponding coefficients must be equal. If a+ b+ c / a+ b+ c for all real then a = a, b = b and c = c Proof If a+ b+ c = a+ b+ c for more than two values of, then ( a  a ) + ( b  b ) + ( c  c ) = 0. That is, a = a, b = b and c = c. EXAMPLES. Write in the form A (  ) + B (  ) + C. A ]  g + B (  ) + C= A (  + ) + B B+ C = A  A + A + B  B + C = A + ( A + B) + A  B + C For / A + ( A + B) + A  B + C A = ( )  A + B = 3 ( ) A  B + C = 5 ( 3) Substitute ( ) into ( ):  ( ) + B = B = 3 B = Substitute A = and B = into ( 3):  + C = 5 + C = 5 C = 4 ` / (  ) + (  ) + 4 You learnt how to solve simultaneous equations with 3 unknowns in Chapter 3. CONTINUED
23 564 Maths In Focus Mathematics Etension Preliminar Course. Find values for a, b and c if  / a( + 3) + b + c . a ] + 3g + b + c  = a ( ) + b + c  = a + 6a + 9a + b + c  = a + ( 6a + b) + 9a + c  For  / a + ( 6a + b) + 9a + c  a = ( ) 6a + b =  ( ) 9a + c  = 0 ( 3) Substitute ( ) into ( ): 6 ( ) + b = b =  b =7 Substitute ( ) into ( 3): 9 ( ) + c  = c = 0 c =8 ` a =, b =  7, c = Find the equation of the parabola that passes through the points (, 3),( 0, 3) and (, ). The parabola has equation in the form = a + b + c. Substitute the points into the equation: ^ ,3h:  3 = a]  g + b]  g+ c ` = a  b + c a  b + c = 3 ] g ^ 0, 3h: 3= a] 0g + b] 0g+ c ` = c c = 3 ] g ^, h: = a] g + b] g+ c = 4a + b + c ` 4a + b + c = ] 3g Solve simultaneous equations to find a, b and c. Substitute () into (): a  b + 3 = 3 a  b = 6 ( 4)
24 Chapter 0 The Quadratic Function 565 Substitute () into (3): 4a + b + 3 = 4a + b = 8 ( 5) ( 4) # : a  b =  ( 6) ( 6) + ( 5): a  b =  4a + b = 8 6a = 6 a = Substitute a = into(5): 4 ( ) + b = b = 8 b = 4 b = 7 ` a =, b = 7, c = 3 Thus the parabola has equation = Eercises. Find values of a, b and c for which (a) / a ] + g + b ] + g + c (b) / a ] + g + b ] + g + c (c)   / a ]  g + b ]  g+ c (d) / a ]  3g + b ]  3g + c (e) / a ] + g + b ]  g+ c (f) / a ]  g + b ]  g + c (g) / a ] + 4g + b ] + g + c (h) / a ] + g + b+ c (i) / a ] + 3g + b ] + 3g+ c (j) / a ]  g + b ] + g+ c. Find values of m, p and q for which   / m ] + g + p ] + g + q. 3. Epress in the form A ]  g+ B ] + g + C Show that can be written in the form a]  g] + 3g + b]  g + c where a =, b = and c = Find values of A, B and C if +  / A]  g + B + C. 6. Find values of a, b and c for which / a ] + 3g + b + c ] + g. 7. Evaluate K, L and M if / K]  3g + L] + g  M.
25 566 Maths In Focus Mathematics Etension Preliminar Course 8. Epress 4 + in the form a ] + 5g + b]  3g + c. 0. Find the equation of the parabola that passes through the points 9. Find the values of a, b and c if (a) (0,  5), (,  3) and ( 3, 7) (b) (,  ), (3, 0) and (, 0) 07 / a]  4g  b] 5 + g+ c. (c) (, ), (, 6) and (, ) (d) (, 3), (,  4) and (,  ) (e) (0, ), (, ) and (,  7) Sum and Product of Roots When ou solve a quadratic equation, ou ma notice a relationship between the roots. You also used this to factorise trinomials in Chapter. EXAMPLE (a) Solve = 0. (b) Find the sum of the roots. (c) Find the product of the roots. Notice 9  is the coefficient of and 0 is the constant term in the equation. (a) = 0 (  4)(  5) = 04 = 0,  5 = 0 ` = 4, = 5 (b) Sum = = 9 (c) Product = 4# 5 = 0 This relationship with the sum and product of the roots works for an quadratic equation. The general quadratic equation can be written in the form ( a + b) + ab = 0 where a and b are the roots of the equation. Proof Suppose the general quadratic equation a + b + c = 0 has roots a and b. Then this equation can be written in the form
26 Chapter 0 The Quadratic Function 567 (  a)(  b) = 0 i.e.  b  a + ab = 0 ( a + b) + ab = 0 EXAMPLES. Find the quadratic equation that has roots 6 and . Method : Using the general formula ( a + b) + ab = 0 where a = 6 and b = a + b = 6+  = 5 ab = 6#  =6 Substituting into ( a + b) + ab = 0 gives = 0 It doesn t matter which wa around we name these roots. Method : If 6 and  are the roots of the equation then it can be written as ]  6g] + g = = = 0. Find the quadratic equation that has roots 3 + and 3 . Method : Using the general formula a + b = = 6 ab = ( 3 + ) # ( 3  ) = 3 ( ) = 9  = 7 Substituting into ( a + b) + ab = 0 gives = 0 Method : If 3 + and 3  are the roots of the equation then it can be written as _ " 3 +, i_ " 3 , i = 0 ^ h^ h = = = 0
27 568 Maths In Focus Mathematics Etension Preliminar Course We can find a more general relationship between the sum and product of roots of a quadratic equation. If a and b are the roots of the quadratic equation a + b + c = 0: b Sum of roots: a + b =  a c Product of roots: ab = a Proof If an equation has roots a and b, it can be written as ( a + b) + ab = 0. But we know that a and b are the roots of the quadratic equation a + b + c = 0. Using quadratic identities, we can compare the two forms of the equation. a + b + c = 0 a b c 0 a + a + a = a b c + a + a = 0 b c For ( a + b) + ab / + a + a b ( a + b) = a b ` a + b =  Also ab = c a a EXAMPLES. Find (a) a + b (b) ab (c) a = 0. + b if a and b are the roots of (a) b a + b =  a ( 6) =  = 3
28 Chapter 0 The Quadratic Function 569 (b) ab = = c a (c) a + b! ( a + b) ^ a + bh = a + ab + b ^ a + bh  ab = a + b ] 3g  c m = a + b 9  = a + b 8 = a + b. Find the value of k if one root of k k + = 0 is . If  is a root of the equation then = satisfies the equation. Substitute = into the equation: k] g  7]  g+ k + = 0 4k k + = 0 5k + 5 = 0 5k =5 k =3 3. Evaluate p if one root of +  5p = 0 is double the other root. You could use b and b instead. If one root is a then the other root is a. Sum of roots: b a + b =  a a + a =  3a = a = 3 CONTINUED
29 570 Maths In Focus Mathematics Etension Preliminar Course Product of roots: c ab = a 5p a # a =  a =5p c m = 5p 3 4 c m =5p 9 8 =5p = p Eercises Reciprocals are n and n.. Find a + b and ab if a and b are the roots of (a) + + = 0 (b) = 0 (c) = 0 (d) = 0 (e) = 0. If a and b are the roots of the quadratic equation = 0, find the value of (a) a + b (b) ab (c) + a b (d) a + b 3. Find the quadratic equation whose roots are (a) and  5 (b)  3 and 7 (c)  and  4 (d) and 45 (e) + 7 and Find the value of m in + m  6 = 0 if one of the roots is. 5. If one of the roots of the quadratic equation k  = 0 is  3, find the value of k. 6. One root of 3  (3b + ) + 4b = 0 is 8. Find the value of b. 7. In the quadratic equation k = 0, one root is double the other. Find the value of k. 8. In the quadratic equation p  = 0, one root is triple the other. Find the value of p. 9. In the quadratic equation ( k  ) k + 3 = 0, the roots are reciprocals of each other. Find the value of k.
30 Chapter 0 The Quadratic Function In the quadratic equation + m + = 0, the roots are consecutive. Find the values of m.. In the quadratic equation 3 ( k + ) + 5 = 0, the roots are equal in magnitude but opposite in sign. Find the value of k.. Find values of n in the equation 5( n  ) + = 0 if the two roots are consecutive. 3. If the sum of the roots of + p + r = 0 is  and the product of roots is  7, find the values of p and r. 4. One root of the quadratic equation + b + c = 0 is 4 and the product of the roots is 8. Find the values of b and c. 5. The roots of the quadratic equation a = 0 are b + and b  3. Find the values of a and b. 6. Show that the roots of the quadratic equation 3m + + 3m = 0 are alwas reciprocals of one another. 7. Find values of k in the equation k + ( k + ) + c + m = 0 if: 4 (a) roots are equal in magnitude but opposite in sign (b) roots are equal (c) one root is (d) roots are reciprocals of one another (e) roots are real. 8. Find eact values of p in the equation + p + 3 = 0 if (a) the roots are equal (b) it has real roots (c) one root is double the other. 9. Find values of k in the equation + k + k  = 0 if (a) the roots are equal (b) one root is 4 (c) the roots are reciprocals of one another. 0. Find values of m in the equation m + + m  3 = 0 if (a) one root is  (b) it has no real roots (c) the product of the roots is. Consecutive numbers are numbers that follow each other in order, such as 3 and 4. Equations Reducible to Quadratics To solve a quadratic equation such as ]  3g ]  3g  = 0, ou could epand the brackets and then solve the equation. However, in this section ou will learn a different wa to solve this. There are other equations that do not look like quadratic equations that can also be solved this wa.
31 57 Maths In Focus Mathematics Etension Preliminar Course EXAMPLES. Solve ] + g  3] + g  4 = 0. Let u = + Then u  3u  4 = 0 ] u  4g] u + g = 0 u  4 = 0, u + = 0 u = 4, u =  But u = + So + = 4, + =  =, = 3. Solve + = 3 where! 0. + = 3 # + # = 3 # + = = 0 ]  g]  g = 0  = 0,  = 0 =, = 3. Solve = 0. 9 = ^3 h = ^3 h So = 0 can be written as ^3 h = 0 Let k = 3 k  4k + 3= 0 ] k  3g] k  g = 0 k  3 = 0, k  = 0 k =, k = 3 But k = 3 So 3 =, 3 = 3 = 0, =
32 Chapter 0 The Quadratic Function Solve sin + sin  = 0 for 0c # # 360c. Let sin = u Then u + u  = 0 ] u  g] u + g = 0 u  = 0or u + = 0 u = u =  u = But u = sin So sin = or sin = sin = has solutions in the st and nd quadrants sin 30c = So = 30c, 80c 30c = 30c, 50c For sin =, we use the graph of = sin 60c 30c 3 See Chapter 6 if ou have forgotten how to solve a trigonometric equation. 90c 80c 70c 360c  From the graph: = 70c So solutions to sin + sin  = 0 are = 30c, 50c, 70c CONTINUED
33 574 Maths In Focus Mathematics Etension Preliminar Course 0.6 Eercises. Solve (a) ]  g + 7]  g + 0 = 0 (b) ^  3h ^  3h  = 0 (c) ] + g  ] + g  8 = 0 (d) ] n  5g + 7] n  5g + 6 = 0 (e) ] a  4g + 6] a  4g  7 = 0 (f) ^p + h  9^p + h + 0 = 0 (g) ] + 3g  4] + 3g  5 = 0 (h) ] k 8g  ] k 8g  = 0 (i) ] t  g + ] t  g  4 = 0 (j) ] b + 9g  ] b + 9g  5 = 0. Solve (! 0). 6 (a)  = 6 (b) + = 5 0 (c) = 0 5 (d) + = 8 (e) + = 3. Solve (a) = 0 (b) = 0, giving eact values (c) ^  h + ^  h  = 0 giving eact values (d) ^ + 3h  7^ + 3 h + 0= 0 correct to decimal places (e) ^ a + 4ah + ^a + 4ah  8 = 0 giving eact values. 4. Solve (a) = 0 (b) 3 p p = 0 (c) = 0 (d) = 0 (e) = Solve + = 5(! 0). 6. Solve b + l + b + l  = 0 (! 0). 7. Solve d + n  9d + n + 0 = 0 correct to decimal places (! 0). 8. Solve for 0c # # 360c. (a) sin  sin = 0 (b) cos + cos = 0 (c) sin sin  = 0 (d) cos = cos (e) sin = cos  9. Solve for 0c # # 360c. (a) tan  tan = 0 (b) cos  = 0 (c) sin  sin = 0 4 (d) 8sin  0sin + 3 = 0 4 (e) 3tan  0tan + 3 = 0 0. Show that the equation = 5 has real + 3 irrational roots (!  3).
34 Chapter 0 The Quadratic Function 575 Test Yourself 0. Solve (a)  3 # 0 (b) n  90 (c) 4  $ 0. Evaluate a, b and c if = a ( + ) + b( + ) + c. 3. Find (a) the equation of the ais of smmetr and (b) the minimum value of the parabola = Show that = is a positive definite quadratic function. 5. If a and b are roots of the quadratic equation = 0, find (a) a + b (b) ab (c) + a b (d) ab + a b (e) a + b 6. Solve (3  )  (3  )  3 = Describe the roots of each quadratic equation as (i) real, different and rational (ii) real, different and irrational (iii) equal or (iv) unreal. (a) = 0 (b) = 0 (c) = 0 (d) = 0 (e) = 0 9. Find (a) the equation of the ais of smmetr and (b) the maimum value of the quadratic function = Write in the form a (  ) + b ( + 3) + c.. Solve sin + sin  = 0 for 0c# # 360c.. Find the value of k in k  = 0 if the quadratic equation has (a) equal roots (b) one root  3 (c) one root double the other (d) consecutive roots (e) reciprocal roots Solve = 5 + (! 0). 4. Find values of m such that m for all. 5. Solve = For each set of graphs, state whether the have (i) points (ii) point (iii) no points of intersection. (a) = 7 and = 0 (b) + = 9 and = 33 (c) + = and = 0 (d) = and = 3 + (e) = and = Show that for all.
35 576 Maths In Focus Mathematics Etension Preliminar Course 7. State if each quadratic function is (i) indefinite (ii) positive definite or (iii) negative definite. (a) (d) 8. Show that k  p + k = 0 has reciprocal roots for all. (b) 9. Find the quadratic equation that has roots (a) 4 and  7 (b) and Solve = 0. (c). Solve 3 (a) 7 + n (b) $ n  3 (c) (d) (e) # $ 54 Challenge Eercise 0. Show that the quadratic equation  k + k  = 0 has real rational roots.. Find the equation of a quadratic function that passes through the points (8, ),( 3,  ) and (, 0 ). 3. Find the value of a, b and c if / a( + ) + b( + ) + c Solve + + = Find the maimum value of the function f ( ) = Find the value of n for which the equation ( n + ) = 0 has one root triple the other.
36 Chapter 0 The Quadratic Function Find the values of p for which  + 3p  0 for all. 8. Show that the quadratic equation  p + p = 0 has equal roots. 9. Solve = Find values of A, B and C if / ( A + 4) + B( + 4) + C Epress   in the form a b Find eact values of k for which + k + k + 5 = 0 has real roots. 3. Solve 3  cos  3sin = 0 for 0c # # 360 c. 4. Solve b + l  5b + l + 6 = Solve sin + cos  = 0 for 0c # # 360 c. 6. If a and b are the roots of the quadratic equation = 0, evaluate 3 3 a + b.
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