2.Quadratic Equations

Quadratic Equations

CHAPTER 2: QUADRATIC EQUATIONS

1.1 Recognize Quadratic Equations and express it in general form

General form

bx + c = 0 , where a , b and c are constants , a 0

Properties1. Equation must be in one unknown only2. The highest power of the unknown is 2

Examples1.2x 2 + 3x – 1 = 0 is a quadratic equation2.4x 2 – 9 = 0 is a quadratic equation3.8x 3 – 4x2 = 0 is not a quadratic equation

Activity 1

1. Determine whether each of the following equation is a quadratic equation or not . Equations Answer

(a) x 2 – x = 0 Yes

(b) 2x 2 – y = 0

(c) 3x + 2 = 0

(d) 2m 2 – 7m – 3 = 0

(e) k 2 – 4k = 0

(f) y 2 – 2 = 0

2. Rewrite each of the following quadratic equation in the general form. State the value of a , b and c .

Quadratic equations Value of a , b and c(a) 1 + 2x = x(x + 3) 1 + 2x = x2 + 3x x2 + x – 1 = 0

a = 1b = 1 c = -1

(b) m 2 = 21 – 4m

PPMP Matematik TambahanNegeri Johor

Quadratic Equations

(c) (y + 6)(y – 2) = - 7

(d) x 2 =

(e) (x + 1) 2 = 16

1.2 Roots of Quadratic Equations

Notes1. The root of a quadratic equation is the value(number) of the unknown(variable) that satisfy the equation .2. A quadratic equation has at most two roots only

Exercises

1. Determine which of the values of the variable x given are roots of the respective quadratic equation.

(a) x 2 – x – 2 = 0 ; x = - 1 , 1 , 2 (b) 2x 2 + 7x + 3 = 0 ; x = - 3, - , 1 , 3

2. Determine by inspection which of the values of x are roots of the following quadratic equations .

(a) (x + 3)(x – 2) = 0 ; x = 3 , 2 , - 3 (b) x(x + 4) = 0 ; x = 4 , 0 , - 4

3. If x = 2 is the root of the quadratic equation x 2 – 3kx -10 = 0 , find the value of k .


Quadratic Equations

2 . SOLVING QUADRATIC EQUATION

2.1 Solving Quadratic Equations

A. By Factorization If a quadratic equation can be factorized into a product of two factors such that (x – p)(x – q) = 0 , Hence x – p = 0 or x – q = 0 x = p or x = q p and q are the roots of the equation .

Notes 1. If p q the equation have two different roots

2. If p = q the equation have two equal roots (one root only)3. The equation must be written in general form ax 2 + bx + c = 0 before

factorization.

Activity 2Solve the following quadratic equations by factorization .1. x 2 – 7x – 8 = 0 ( x – 8 ) ( x + 1 ) = 0 x – 8 = 0 or x + 1 = 0 x = 8 or x = -1

2. x 2 – 4x + 4 = 0

3. x 2 – 8x = 0 4. 4x 2 – 9 = 0

5. 6x 2 + 13x – 5 = 0 6. (3x + 1)(x - 1) = 7

7. 8. (x + 1)(x – 5) = 16 x2 – 4x – 5 = 16 x2 – 4x – 21 = 0 ( x – 7 ) ( x – 3 ) = 0 x = 7 or x = 3


Quadratic Equations

9. 10. (2p + 1)(p + 1) = 0

Exercise 1Solve the following quadratic equation by factorisation.1. x 2 – 5x – 6 = 0 [6,-1] 9. x 2 – 9x + 20 = 0 [5,4]

2. m 2 + 5m – 24 = 0 [-8,3]

10. 4x 2 – 13x + 3 = 0 [ ]

3. y 2 + 10y + 24 = 0 [-6,-4] 11. 2x 2 – 3 = 5x [ ]

4. 2x 2 + 3x – 5 = 0 [1, ] 12. 6x 2 – 11x = 7 [ ]

5. 16x 2 – 6x – 7 = 0 [ ] 13. (2x – 3) 2 = 49 [ 5,-

2]

6. 2a 2 + 4a = 0 [0.-2] 14. (3m + 1)(m – 1) = 7 [ ]

7. 100 – 9n 2 = 0 [ ] 15. 10x 2 + 4 = 13x [

]

8. (2x + 1)(x + 3 ) = 0 [ ] 16. x(x + 4) = 21 [ -7,3]

B. By Completing the Square Notes 1. The expression x 2 – 2x + 1 can be written in the form (x – 1) 2 This is called “perfect square”.

Example

Solve each of the following quadratic equation (a) (x + 1) 2 = 9 (b) x 2 = 49 x + 1 = 3 x + 1 = 3 , x + 1 = -3 x = 2 , x = - 4


Quadratic Equations

(c) (x + 2) 2 = 36

2. From the example , note that, if the algebraic expression on the LHS of the quadratic equation are perfect squares , the roots can be easily obtained by finding the square roots.

3. To make any quadratic expression x2 + hx into a perfect square , we add the term

( to the expression .

And this will make

4. To solve the equation by using completing the square method for quadratic equation ax 2 + hx + k = 0 , follow this steps ;

Step 1 : Rewrite the equation in the form ax 2 + hx = - k Step 2 : If the coefficient of x2 is 1 , reduce the coefficient to 1 (by dividing) .

Step 3 : Add ( )2 to both sides of the equation.

Step 4 : Write the expression on the LHS as perfect square. Step 5 : Solve the equation

Examples1. x2 + 6x – 9 = 0 2. 2x2 – 5x – 8 = 0 x2 + 6x = 9

x2 + 6x + = 9 +

( x + 3 )2 = 18 x + 3 = x + 3 = 4.243 x = 4.243 – 3 , x = -4.243 – 3 x = 1.243 , x = -7.243

Exercise 2 Solve the following equations by completing the square. (Give your answers correct to four significant figures)

1. x 2 – 8x + 14 = 0 [5.41 , 2.59]


Quadratic Equations

2. 2x 2 – 7x – 1 = 0 [3.64 , -0.14]3. x 2 + 5x + 1 = 0 [-0.209,-4.79]4. – x 2 – 3x + 5 = 0 [-4.19,1.19]5. x 2 = 5(x + 4) [7.62 , -2.62]6. -4x 2 – 12x + 3 = 0 [-3.23,0.232]7. 2x 2 – 3x – 4 = 0 [2.35,-0.85]

C. By Using the quadratic formula The quadratic equation ax 2 + bx + c can be solved by using the quadratic formula

x = , where a 0

Example 2x 2 – 7x – 3 = 0 a = 2 , b = -7 , c = -3

x = 3.886 , -0.386

Exercise 3

Use the quadratic formula to find the solutions of the following equations. Give your answers correct to three decimal places .

1. x 2 – 3x – 5 = 0 [4.193 , -1.193]2. 9x 2 = 24x – 16 [1.333 ]3. 2x 2 + 5x – 1 = 0 [0.186 , -2.686]4. 3x 2 + 14x – 9 = 0 [2.899 , -6.899]5. 7 + 5x – x 2 = 0 [0.768 , -0.434]6. m 2 = 20 – 4m [0.573 , -5.239]

7. [-1.140 , 6.140]

8. x(x + 4) = 3 [0.646 , -4.646]


Quadratic Equations

2.2 Forming a quadratic equation from given roots

A. If the roots of a quadratic equation are known, such as x = p and x = q then, the quadratic equation is (x – p)(x – q) = 0 x 2 – px – qx + pq = 0 x 2 – (p + q)x + pq = 0

Notice that p + q = sum of roots ( SOR ) and pq = product of roots ( POR )

Hence, the quadratic equation with two given roots can be obtained as follows :-

x 2 – (SOR)x + (POR) = 0

ExamplesForm the quadratic equations from the given roots.1. x = 1 , x = 2

Method 1 Method 2 (x – 1)(x – 2) = 0 SOR = 1 + 2 = 3 x2 - 2x – x + 2 = 0 POR = 1 x 2 = 2 x2 - 3x + 2 = 0 x2 – 3x +2 = 0

2. x = - 2 , x = 3

Exercise 4

Form the quadratic equations with the given roots.

1. x = 3 , x = 2 [x2 - 5x + 6 = 0]

2. x = - 6 , [3x2 +17x - 6 = 0 ]

3. x = - 4 , x = - 6 [x2 + 10x + 24 = 0]


Quadratic Equations

4. x = -3 , x = [5x2 + 11x - 12=0 ]

5. x = -7 , 3 [x2 + 4x - 21 = 0]

6. x = 5 only [x2 - 10x + 25 = 0]

7. x = 0 , x = [3x2 - x = 0]

8. [6x2 - 5x + 1 = 0]

B. To find the S.O.R and P.O.R from the quadratic equation in general form

ax 2 + bx + c = 0

a , x 2 + = 0

Compare with x 2 – (SOR)x + (POR) = 0

Then , SOR =

POR =

Activity 31. The roots for each of the following quadratic equations are and . Find the value of + and for the following equation

Quadratic Equationsa. x 2 – 12x + 4 = 0 12 4


If and are the roots of the quadratic equation ax2 + bx + c = 0,

then + =

=

Quadratic Equations

b. x 2 = 4x + 8

c. 3 – 2x 2 = 10x

d. 3x 2 + 8x = 10

e. 2x 2 + 3x + 4 = 0

C. Solving problems involving SOR and POR

Activity 4

1. Given that and are the roots of the quadratic equation 2x 2 + 3x + 4 = 0 . Form a quadratic equation with roots 2 and 2.

2x 2 + 3x + 4 = 0 New roots

SOR = = 2 ( ) = = -3

= POR = 4 = 4(2) = 8

= 2 x 2 – (SOR)x + (POR) = 0 x 2 – (-3)x + 8 = 0 x 2 + 3x + 8 = 0


Quadratic Equations

2. If and are the roots of the quadratic equation 2x 2 – 5x – 1 = 0 , form a quadratic equation with roots 3 and 3.

3. Given that and are the roots of the quadratic equation 2x 2 – 3x + 4 = 0 . Form a

quadratic equation with roots and .

4. Given that m and n are roots of the quadratic equation 2x2 – 3x – 5 = 0 , form a quadratic

equation which has the roots and .


Quadratic Equations

Exercise 5

1. If and are roots of the quadratic equation 2x2 + 3x + 1 = 0, form a quadratic equation for the following rootsa. 2 and 2 [x2 + 3x + 2 = 0]

b. 2 + 3 and 2 + 3 [x2 - 3x + 2 = 0 ]

c. and [8x2 + 6x + 1 = 0 ]

d. 2 - 1 and 2 - 1 [x2 - 6x - 5 = 0]

2. If and are the roots of equation 2x 2 – 5x – 6 = 0 , form a quadratic

equation with roots and . [ ]

3. Given that and are the roots of the equation 3x 2 = 4 – 9x , form a quadratic equation with roots and . [ ]

4. Given m and n are the roots of the equation x 2 + 10x – 2 = 0 , form a quadratic equation with roots;

(a) 2m + 1 and 2n + 1 [ ]

(b) and [ ]

5. Given that and are the roots of the equation x 2 + 2bx + 3a = 0 , prove that 4a = b 2 .

6. Given one of the root of the quadratic equation x 2 – 5kx + k = 0 is four times the

other root, find the value of k . [ ]

7. One of the roots of the quadratic equation 2x2 + 6x = 2k – 1 is twice the value of the other root whereby k is a constant. Find the roots and the value of k.

[-1, -2 ; k = ]


Quadratic Equations

3. DISCRIMINANT OF A QUADRATIC EQUATIONS 3.1 Determining the types of roots of quadratic equations

For the quadratic equation ax2 + bx + c = 0 , the value of b2 – 4ac will determine the types of roots.

b2 – 4ac is called the “discriminant”

Condition Type of rootsb2 – 4ac > 0 Two different rootsb2 – 4ac = 0 Two equal rootsb2 – 4ac < 0 No roots

ExampleDetermine the type of roots for each of the following quadratic equations .

(a) 2x2 – 7x + 9 = 0 (b) 2x2 – 3x – 9 = 0

a = 2 , b = -7 , c = 9 b2 – 4ac = (-7)2 – 4(2)(9) = 49 – 72 = -23 < 0 no roots

Exercise 6

Calculate the discriminant for each of the following quadratic equation and then state the type of roots for each equation .

1. x2 – 8x + 14 = 0 5. x(3x – 5) = 2x- 5

2. 2x2 – 7x – 1 = 0 6. 5(5 – 4x) = 4x2


Quadratic Equations

3. 4 + x2 = 4x 7. x2 = 2 – 4x

4. (x – 2)2 = 3 8. 2x2 + 3x = 0

3.2 Solving problems involving the use of the discriminant

Activity 5

1. The quadratic equation 2kx2 + 4x – 3 = 0 has two equal roots , find the value of k .

2. The quadratic equation x2 + 2kx + (k + 1)2 = 0 has real roots , find the range of values of k.


Quadratic Equations

3. Show that the equation x2 + m + 1 = 8x has two different roots if m < 15 .

4. The straight line y = tx – 2 is a tangent to the graph of a curve y = 2x2 + 4x , find the value of t (t > 0) .

5. Given that the quadratic equation p(x2 + 9) = - 5qx has two equal roots , find the ratio of p : q . Hence, solve those quadratic equation .

6. Show that the quadratic equation x2 + kx = 9 – 3k has real roots for all the value of k .


Quadratic Equations

Exercise 7

1. Find the possible values of m if the quadratic equation (4 – 2m)x2 – 2m = 1 – 3mx has two equal roots .

2. The equation x2 – 2x + = 0 has two different roots , find the range of values of k .

3. Given that the equation (p + 1)x2 – 2x + 5 = 0 has no roots , find the range of values of p .

4. Find the range value of k if the quadratic equation x2 + 1 = k – 4x has real roots .

5. The quadratic equation 2x(x – 3) = k – 2x has two distinct roots. Find the range of values of k.

6. The quadratic equation (m – 2)x 2 + 2x + 3 = 0 has two distinct roots. Find the range of values of m.

7. A quadratic equation 4x(x + 1) = 4x – 5mx – 1 has two equal roots. Find the possible values of m.

8. The straight line y = 2x – 1 does not intersect the curve y = 2x 2 + 3x + p. Find the range of values of p.

9. The straight line y = 6x + m does not intersect the curve y = 5 + 4x – x 2 . Find the range of values of m.10. The straight line y = 2x + c intersect the curve y = x2 – x + 1 at two different points, find the range of values of c.

11. Find the range values of m if the straight line y = mx + 1 does not meet the curve y2 = 4x .

12. Show that the quadratic equation kx2 + 2(x + 1) = k has real roots for all the values of k.

Answers for Exercise 7

1. 2. k < 3 3. p >

4. 5. k > -2 6. m <


Quadratic Equations

7. m = or m = 8. p > –

9. m > 6 10. . c > 11. m > 1

Enrichment Exercise – Quadratic Equations

1. The quadratic equation kx2 + 4x + 3= 0 has two different roots, find the range of values of k .

2. Find the possible values of k if the quadratic equation x2 + (2 + k)x + 2(2 + k) = 0 has two equal roots.

3. Show that the quadratic equation x2 + (2k – 1)x + k2 = 0 has real roots if k .

4. Find the possible values of k if the straight line y = 2x + k is a tangent to the curve y = x2 + x + 1 .

5. Given that and are the roots of the quadratic equation 2x2 – 8x + 1 = 0 . Form the quadratic equation with roots and .

6. Solve each of the following quadratic equation :- a. 6x2 + 5x – 4 = 0 b. y(y + 1) = 10 c. 2x(x + 5) = 7x + 2 d. 16x2 + 8x + 1 = 0

7. The roots of the equation 2ax2 + x + 3b = 0 are and . Find the value of a and b.

8. If and are the roots of quadratic equation 2x2 – 3x – 6 = 0 , form the quadratic

equation with roots and .

9. Given and – 5 are the roots of the quadratic equation . Write the quadratic equation

in the form of ax2 + bx + c = 0 .

10. Given that m + 2 and n – 1 are the roots of the equation x2 + 5x = - 4 . Find the possible values of m and n .

11. Given that 2 and m are the roots of the equation (2x – 1)(x + 3) = k(x – 1) such that


Quadratic Equations

k is a constant . Find the value of m and k .

12. Given one of the root of the equation 2x2 + 6x = 2k – 1 is twice the other root, such that k is a constant . Find the value of the roots and the value of k .

13. One of the root of the quadratic equation h + 2x – x2 = 0 is - 1 . Find the value of h.

14. Form the quadratic equation which has the roots -3 and . Give your answer in the

form ax2 + bx + c = 0 , where a , b and c are constants. (SPM 2004)

15. Solve the quadratic equation x(2x - 5) = 2x – 1 . Give your answer correct to three decimal places .(SPM 2005)

16. The straight line y = 5x – 1 does not intersect the curve y = 2x2 + x + p . Find the range of the values of p .(SPM 2005)

17. A quadratic equation x2 + px + 9 = 2x has two equal roots. Find the possible values of p.(SPM 2006)

Answers on Enrichment Exercises

1. k < 2. k = 6 , - 2

4. k = 5.

6. (a) (b) y = 2.702 , - 3.702 (c) (d)

7. a = 3 , b = -4 8. 9. 10. n = 0 , - 3 ; m = - 6 , - 3

11. m = 3 , k = 15 12. roots = - 1 , -2 and k =

13. h = 3 14. 2x2 + 5x – 3 = 0

15. x = 3.35 , 0.15 16. p > 1

17. p = -8 , 4


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2.Quadratic Equations