Quadratic Theory

Higher Maths

Quadratic TheoryThe quadratic graph

Using the discriminant

Quadratic theory examples

Basic skills questions

Problem solving questions

Past paper questions Ans

Ans

Ans

Click on a topic

In each of the diagrams below state whether (i) a>0 or a<0

(ii) b2-4ac<0 or b2-4ac>0 or b2-4ac=0

The quadratic graph y = ax2+bx +c

Continued on next slide

Using the Discriminant

In the solution of the quadratic equation ax2 + bx + c = 0

the solutions are given by

a

acbbx

2

)4( 2

The quantity b2 - 4ac is important .

It is called the discriminant.

It can be used to tell what kind of roots the equation will have.

The table on the next slide investigates this idea.

Continued on next slide

Consider the following table. You are required to complete the table

10. f(x) = -3x2 - x + 2

9. f(x) = 5x2 + 3x - 2

8. f(x) = -2x2 + 3x +1

7. f(x) = x2 + 4x + 4

6. f(x) = 2x2 + x + 6

5. f(x) = x2 - 7x - 2

4. f(x) = 3x2 + x - 4

3. f(x) = x2 - 6x + 9

2. f(x) = x2 + 2x + 5

65 -7 3 2 21. f(x) = 2x2 + 3x - 7

c b aNumber of roots f(x) = ax2 + bx + c

See next slide before starting table

You could use the graphic calculator for this work if you wish.

Enter the function in Y1.Press ZOOM and select 6: ZStandard

Draw each graph and determine the number of roots the quadratic has.

Enter the values of a , b and c and evaluate .

Examine the table carefully and make some conjectures about the connection between

the value of b2 - 4ac and the number of roots of the corresponding quadratic equation.

Quadratic Theory Examples [ y = ax2+bx +c ]

1. Choose one of either a > 0 or a < 0

and one of b2 – 4ac > 0 b2 – 4ac = 0 b2 – 4ac < 0

corresponding to each of the six graphs below.

Continued on next slide

2. Use the discriminant b2 – 4ac to find the nature of the roots of the equations below.

a) 2x2 – 7x + 1 = 0 b) 5x2 + 2x + 2 = 0c) 9x2 – 24x + 16 = 0 d) x2 + x + 7 = 0e) 6x2 – x – 1 = 0 f) 3x2 + 2x + 5 = 0

3. Examine the discriminant to see if the roots of the following equations are real, equal or imaginary.

a) 2x2 – 5x – 1 = 0 b) x2 + x + 7 = 0c) 3x2 – 18x + 27 = 0 d) 2x2 + x + 1 = 0

4. Find k given that each of the following equations has equal roots.a) x2 – 8x + k = 0b) kx2 – 12x + 9 = 0

c) x2 + kx + 16 = 0

5. Find m if x2 + 2mx + 9 = 0 has equal roots.

Continued on next slide

9

1

5x2

10x4x 2

6. Find p if x2 + (p + 1)x + 9 = 0 has real distinct roots.

7. Find p if (p + 1)x2 – 2(p + 3)x + 3p = 0 has equal roots.

8. find c if x2 + (x + c)2 = 8 has equal roots.

9. Show that the roots of k(x + 1)(x + 4) = x are not real if

10. Find m if x2 + (mx – 5)2 = 9 has equal roots.

11. If = n form a quadratic equation in x and show

that, for real x, n – 3 or n 2.

< k < 1

Solutions on next slide

Quadratics Solutions

1. i) a < 0 b2 – 4ac > 0 ii) a > 0 b2 – 4ac < 0iii) a > 0 b2 – 4ac = 0 iv) a < 0 b2 – 4ac = 0v) a > 0 b2 – 4ac 0 vi) a < 0 b2 – 4ac < 0

2. a) = 41 roots are real and distinctb) = – 36, no rootsc) = 0, roots are equald) = – 27, no rootse) = 25 , roots are real and distinctf) = – 56, no roots

Continued on next slide

3. a) (–5)2 – 4.2.(–1) = 33 real rootsb) 12 – 4.1.7 = – 27 imaginary rootsc) (–18)2 – 4.3.27 = 0 equal rootsd) 12 – 3.2.1 = – 7 imaginary roots

4. a) (–8)2 – 4.1.k = 0 , 64 – 4k = 0 , k = 16b) (–12)2 – 4.k.9 = 0 , 144 – 36k = 0 , k = 4c) k2 – 4.1.16 = 0 , k2 = 64 , k = 8

5. (2m)2 – 4.1.9 = 0 , 4m2 = 36 , m2 = 9 , m = 3

6. (p + 1)2 – 4.1.9 > 0 , (p + 1)2 – 62 > 0 , (p – 5)(p + 7) > 0 p < – 7 or p > 5

7. [–2(p + 3)]2 – 4(p + 1)(3p) = 04(p2 + 6p + 9) – 12p2 – 12p = 04p2 + 24p + 36 – 12p2 – 12p = 0

2p2 – 3p – 9 = 0(p – 3)(2p + 3) = 0p = – 3/2 or p = 3

Continued on next slide

8. x2 + (x + c)2 = 82x2 + 2cx + c2 – 8 = 0

This has equal roots when (2c)2 – 4(2)(c2 – 8) = 04c2 –8c2 + 64 = 04c2 = 64c = 4

9. k(x + 1)(x + 4) = xkx2 + (5k – 1)x + 4k = 0

This has imaginary roots if (5k – 1)2 – 4(k)(4k) < 025k2 – 10k + 1 < 09k2 – 10k + 1 < 0(k – 1)(9k – 1) < 0 < k < 1

9

1

Continued on next slide

10. x2 + (mx – 5)2 = 9(m2 + 1) – 10mx + 16 = 0

This has equal roots when (–10m)2 – 4(m2 + 1)(16) = 0100m2 – 64m2 – 64 = 036m2 – 64 = 09m2 = 16m = 4/3

11. x2 + 4x + 10 = n(2x + 5)x2 + (4 –2n)x + 10 – 5n = 0

This has real roots when (4 – 2n)2 – 4(1)(10 – 5n) 016 – 16n + 4n2 – 40 + 20n o4n2 + 4n – 24 on2 + n – 6 0(n + 3)(n – 2) 0n – 3 or n 2

Quadratic theory - Basic skills questions

Basic skills - Solutions

Quadratic Theory – Problem solving questions

Quadratic Theory Exam Level Questions/ Past Paper questions. 1. For what values of ‘p’ does the equation x2 – 2x + p = 0 have equal roots.

2. Show that the roots of the quadratic (k-2)x2 – (3k-2)x + 2k = 0 are always real.

3. If ‘k’ is a real number show that the roots of the equation kx2 + 3x + 3 = k are always real.

4. The roots of the equation (x+1)(x+k) = -4 are equal. Find the value of ‘k’.

5. Find the values of ‘k’ for which the equation 2x2 + 4x – k = 0 has equal roots.

6. Calculate the least positive integer ‘k’so that the graph shown does not cutor touch the x axis.

y = kx2 -8x + k(0,k)

7. Show that the equation (1-2k)x2 – 5kx - 2k = 0 has real roots for all integer values of ‘k’.

8. For what values of ‘k’ has the equation x2 – 5x + (k+6) = 0 have equal roots?

9. If f(x) = 2x+1 and g(x) = x2 + k, show that the equation g(f(x)) – f(g(x)) = 0 reduces to 2x2 + 4x – k = 0 and find the value of ‘k’ for which this equation has equal roots. What kind of roots does this equation have when k=6?

10. For what values of ‘k’ does the equation 5x2 – 2x + k = 0 have real roots?

11.For what value of ‘a’ does the equation ax2 + 20x + 40 = 0 have equal roots?

12. Find ‘p’ given that the equation x2 + (px – 5)2 = 9, has equal roots.

13.Given that , form a quadratic equation in x

and hence show that if or then the roots of the equation are real.

n5x2

10x4x2

3n 2n

14.Find ‘m’ if the equation (2m-1)x2 + (m+1)x + 1 = 0 has equal roots. If m lies between these values find the nature of the roots.

15.Show that the roots of the equation k(x+1)(x+4) = x are not real if

9

1<k<1.

16. Find ‘k’ given that the equation kx2 + (2k+1)x + k = 0 has equal roots.

17. If ‘k’ is a real number, show that the roots of the equation

, are always real.k27x3

21x2x2

18.For what values of ‘k’ does the equation x(x-4) + 2 = k(2x – 3k) have real roots?

19.Show that the line y = x + c meets the parabola y = x2 – 3x where x2 – 4x – c = 0. Find the value of ‘c’ if the line is a tangent to the parabola.

20.Find the value of ‘n’ if the equation , is to have equal roots.

n2x

)2x(2

2

Answers – Exam level questionsQ1. p = 1Q2. (k+2)2 is always greater than or equal to 0 because it is a quantity squared.Q3. (2k-3)2 is always greater than or equal to 0 because it is a quantity squared.Q4. k = 5, k = -3Q5. k = -2Q6. No roots if k<-4 or k>4 therefore the smallest positive integer k is k = 5Q7. b2 – 4ac = 0 when k=0 or k = -8/9 therefore real for all integer values of k.Q8. k = 0.25Q9. k = -2 ; If k = 6 there are two real distinct roots.Q10. k Q11. a = 2.5Q12. p = 4/3 , p = -4/3Q13. n 2 or n -3Q14. m = 1, 5Q15. No roots for <k<1

5

1

9

1

Q16. k = -1/4Q17. Roots are not always real. The roots are real if k 2 or k Q18. Real roots if k is between (1-2) and (1+2)Q19. c = -4Q20. n = 0 and n = 3

9

11