quadratic equations

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ADDITIONAL MATHEMATICS MODULE 3

QUADRATIC EQUATIONS

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CHAPTER 2 : QUADRATIC EQUATIONS MODUL 3

2.1

2.2

CONCEPT MAP

2

GENERAL FORM

3

2.2.1 Identifying Exercises 1

3 3

2.2.2 Recognising general form of quadratic equation .ax2 + bx – c = 0 Exercises 2

4

2.3

SOLVING QUADRATIC EQUATIONS

6

2.3.1

Factorisation Exercises 1

6 6

2.3.2

Completing the square Exercises 1

8 8

4

2.3.3 Quadratic formula Exercises 1

10 10

2.4

PASS YEARS QUESTIONS

12

2.5

ASSESSMENT

13

ANSWERS

15

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CHAPTER 2 : QUADRATIC EQUATIONS MODUL 3 2.1

CONCEPT MAP

QUADRATIC EQUATIONS GENERAL FORM .ax2 + bx + c = 0

Factorization

x    x    = 0

Completing the square x 2     x    = 0

Formula

ROOTS x =  ,

b2 – 4ac

> = 0 (Positive)

=0

< = 0 (negative)

Two different roots

Two Equal roots

No real roots

Types of roots



2.2

GENERAL FORM

2.2.1

Identifying

Example 1

Example 2

4x + 3 = 2x 2x + 3 = 0 The highest power of variable x is 1 Therefore 4x + 3 = 2x is not a quadratic equation

x(x + 5) = 7 x2 + 5x - 7 = 0 The highest power of variable x is 2 Therefore x(x + 5) = 7 is a quadratic equation

Exercises 1 Identify which of the following are quadratic equation 1. 3 =

5 2x

.2.

x(2x + 3) = x - 7

(3m + 5)2 = 8m

3. ( x + 4 )(2x – 6) + 3 = 0

4.

5. x (7 - 2x + 3x2) = 0

6. 3x2 – 5 = 2x( x + 4)



2.2.2 Recognising general form of quadratic equation .ax2 + bx – c = 0

Example 1

Example 1

.x2 = 5x – 9 .x2 – 5x + 9 = 0

4x

Compare with the general form .ax2 + bx – c = 0 Thus, a = 1, b = -5 and c = 9

=

x 2  2x x

4x(x) = x2 – 2x 4x2 - x2 – 2x = 0 3x2 – 2x = 0 Compare with the general form Thus, a = 3, b = - 2 and c = 0

Exercises 2 Express the following equation in general form and state the values of a, b and c 1. 3x =

3.

5 2x

x( x + 4 ) = 3

.2.

(2x + 5) =

7 x

.4. (x – 1)(x + 2) = 3



5.

4 x3 = 5 x x

6.

7. px (2 – x) = x – 4m

9. (7 – 2x + 3x2) =

x2 + px = 2x - 6

8. (2x – 1)(x + 4) = k(x – 1) + 3

x 1 3

10. 7x – 1 =

x 2  2x x



2.3.

SOLVING QUADRATIC EQUATIONS

2.3.1 Factorisation

Example 1

Example 2

.x2 + 6x + 5 = 0 ( x + 3)(x + 2) = 0 .x + 3 = 0 or x + 2 = 0 .x = -3 x =-2

x + 3 .x + 2

3x 2x

.x2 + 6

5x

4(x +3) = x(2x – 1) 4x + 12 = 2x2 - x 2x2 - 5x - 12 = 0 (2x + 3)(x - 4) = 0 2x + 3 = 0 or x - 4 = 0 3 .x = x= 4 2 Therefore, The roots of the equation are 3 .x = and 4 2

Therefore, The roots of the equation are .x = -3 and -2

Exercises 3 Solve the following quadratic equation by factorisation

1.

x2 + 3x - 4 = 0

2.

x2 -2x = 15

3.

4x2 + 4x – 3 = 0

4. 3x2 - 7x + 2 = 0



5.

x2 = 3x – 2

6.

7.

8x2 + x = 21(1 – x)

8. (2y – 1)(y + 4) = -7

9.

4y -

1 =3 y

10.

x(2x - 5) = 12

7  6m =m 3m  2



2.3.2. Completing the square Example 1

Example 2

.x2 – 6x + 7 = 0 .x2 - 6x = -7 2

Rearrange in the form .x2 + px = q

2

6 6 .x – 6x +   =-7 +    2   2  .x2 – 6x + (-3)2 = -7 + (-3)2 Add 2 (x - 3)2 = 2  coefficient..of .x    .x – 3 =  2 2   To both sides .x = 3  2 .x = 3 + 2 or 3 - 2 .x = 4.414 or 1.586 2

2x2 -5x – 1 = 0 2x2 – 5x = 1 Change the 5 1 coefficient 2 .x - x = of x2 to 1 2 2 2 2 5 1  5  5 2 .x - x +   = +  2 2  4   4  2

1 25 5  x  = + 2 16 4  33 = 16 33 5 33 .x - =  = 4 16 4 5 33 5 33 + or 4 4 4 4 5  33 5  33 .x = or 4 4 .x = 2.686 or -0.186

.x =

Exercises 4 Solve the following quadratic equation by completing the square

1. (x + 3 )2 = 16

2. (5x - 4)2 = 24



3. x2 - 8x + 12 = 0

4.

3x2 + 6x – 2 = 0

5.

5x2 – 7x + 1 = 0

6.

2x2 – 3x – 4 = 0

7.

(x + 1)(x - 5) = 4

8.

1 -

1 3 = x 2x 2



Quadratic formula Example 1

Example 2

.x2 + 5x + 2 = 0 .a = 1, b = 5, c = 2

3x2 = 4x + 2 3x2 - 4x – 2 = 0 .a = 3, b = -4 , c = - 2 Using the formula

 b  b 2  4ac Using the formula x = 2a  5  5 2  412  .x = 21

.x =

 5  25  8  5  17 .x = = 2 2  5  17  5  17 .x = or 2 2 = - 0.438 or - 4.562

  4  

 4 23

 43 2 

  4   16  24 6 4  40 = 6 4  40 4  40 .x = or 6 6 = 1.721 and – 0.387

=

Exercises 5 Solve the following quadratic equations by using the quadratic formula

1.

x2 – 11x + 28 = 0

2.

–x2 – 3x + 5 = 0



3. 2x2 + 11x + 9 = 0

4. 3x2 + 14x – 9 = 0

5. 10x(2x – 1) – 8 = x(2x + 35)

6. (x – 1)(4x – 9) + 7 = 10x

7.

11v  2 = 2v v3

8.

x 2  3x  1 =2 x2  x 1



9.

3x  2 x3 + =1 5 x2

2.4

10.

1 x 7 +3= x x5

PAST YEARS QUESTIONS

SPM 2001. PAPER 1 Question 3 1.

Solve the quadratic equation 2x(x + 3) = ( x + 4)(1 - x). Give your answer correct to four singnificant figures.

SPM 2003. PAPER 1 Question 3 1.

Solve the quadratic equasion 2x(x – 4) = (1 – x)(x + 2) Give your answer correct to four significant figures.



2.5 ASSESSMENT ( 30 minutes) 1. Express 3x2 – 2px = 5x - 7p in genegal from

2.

Find the roots of the equation 2x2 + 5x = 12

3. Find the roots of

1 x = , x2 3

4. By using the quadratic formula, solve the equation 2x2 – 5x – 1 = x(4x - 2)



5. Solve the quadratic equation (5x – 3)(x + 1) = x(2x – 5) . Give your answer correct to four significant figures.

6. Given the equation x2 + 4x – 5 = (x – a)2 + b , find the values of a and b



ANSWERS Exercises 1 1. No 3. Yes 5. No

2. Yes 4. Yes 6. Yes

Exercises 2 1. 6x2 – 5 = 0 2. 2x2 + 5x – 7 = 0 3. x2 + 4x – 3 = 0 4. x2 + x – 5 = 0 5. x2 + 7x – 20 = 0 6. x2 + (p – 2)x + 6 = 0 7. px2 + (1 – 2p)x – 4m = 0 8. 2x2 + (7 - k) x + k – 7 = 0 9. 9x2 - 7x + 20 = 0 10. 6x2 + x = 0 Exercises 3 1. 1, -4 1 3 3. , 2 2 5. 1, 2 7 3 , 2 4 1 9. ,1 4

7.

Exercises 4 1. 1, -7 3. 2, 6 5. 0.161, 1.239 7. 5.60, -1.60

a= 6, b = 0, c = -5 a = 2 , b = 5 , c = -7 a = 1, b = 4 , c = - 3 a = 1, b = 1, c = -5 a = 1, b = 7, c = -20 a= 1, b=(p – 2), c= 6 a = p , b = (1 – 2p) , c = -4m a = 2, b = (7 - k), c = (k – 7) a = 9, b = -7 c = 20 a = 6, b = 1, c = 0

2. 5, -3 1 4. , 2 3 3 6. 4, 2 1 7. , -3 2 7 10. 1, 3

2. 4. 6. 8.

1.77, - 0.17 0.457 , - 1.437 2.351 , - 0.851 1.823, -0.823



Exercises 5 1. 4, 7 9 3. -1, 2 5. 2.667, - 0.169 1 7. 2, 2 9. 2.812, - 0.119

2. 1.191, - 4.191 4. 0.573 , -5.239 6. 0.810, 4.94 8. 5.192, -1.925 10. -0.403, -3.069

PAST YEARS QUESTIONS 1. 0.393, -3.393

2. 2.591, -0.2573

ASSESSMENT 1. 3x2 – (2p + 5)x + 7p = 0 3 2. , 4 2 3. – 3, 1 4. - 0.5, -1 5. 0.370, -2.70 6. a = 2, b = -9



SPM

ADDITIONAL MATHEMATICS MODULE 4

QUADRATIC EQUATIONS




CONCEPT MAP

2

2.2 FORMING A QUADRATIC EQUATIONS FROM GIVEN ROOTS  and 

3

2.2.1 Form a quadratic equation with the roots

4

Exercises 1 2.2.2

4

Determine the sum of the roots and product of the roots of the following quadratic equations. Exercises 2

5 5

2.3. TYPES OF ROOTS QUADRATIC EQUATION 2.3.1

2.4 2.4.1

2.4.2

2.4.3

6

Determine the types of roots for each of the following quadratic equations Exercises 3

6

SOLVING PROBLEMS INVOLVING (.b2 - 4ac )

7

Find the values of k for each of the following quadratic equations which has two equal roots

7

Exercises 4

7

Find the range of values of h for each of the following quadratic equations which roots are different

8

Exercises 5

8

Find the range of values of m for each of the following quadratic equations which has no roots

9

Exercises 6

9

2.5


2.6

ASSESSMENT (30 MINUTES)

ANSWERS

10 12 14




CONCEPT MAP QUADRATIC EQUATIONS GENERAL FORM ROOTS .x =  and x = 

FORMING A QUADRATIC EQUATION FROM GIVEN ROOTS

The quadratic equation S Sum of the roots Product of the roots

The quadratic equation

TYPES OF ROOTS



2.2 FORMING A QUADRATIC EQUATIONS FROM GIVEN ROOTS  and 

Notes If x1 = a and x2 = b Then (x – a) = 0 and (x – b) = 0 (x – a)(x – b) = 0 2 .x – bx – ax + ab = 0 .x2 – (a + b)x + ab = 0 Sum of the roots

Product .of the roots

ax2 + bx + c = 0 ax 2 bx c + + =0 a a a .x2 +

bx c + = 0 ……………………(2) a a

Compare with the equations (1) and (2)

–

.x2 – (    )x +  = 0 …….(1)

Sum of the roots

Product .of the roots

From general form

The quadratic equation with roots  and  is written as

(   )=

b c

(   ) = -

 =

.x2 – (    )x +  = 0 ……….….(1) bx c x2 + + = 0 ……………………(2) a a

b c c a

( The sum of the roots)

( The product of the roots)



2.2.1 Form a quadratic equation with the roots

Exercises 1 Sum of .the roots (   )

Roots  and  Example 1 3 , 2

5

Example 2 1 , -3 4

1 + (-3) 4

=

Product of .the roots  6 3 1   (- 3) = 4 4

1  12  11 = 4 4

The puadratic equation .x2 – (    )x +  = 0 x2 – (5)x + 6 = 0   11    3 .x2 –  x +   =0  4   4  4x2 + 11x – 3 = 0

a) 4 , -7

b) 2,

1 3

c) 1 1 , 3 2

d) 1 2 , 5 3

e) 3k,

6k 5



2.2.2

Determine the sum of the roots and product of the roots of the following quadratic equations.

Exercises 2 The puadratic equation

Sum of .the roots

Product of .the roots

Example 1 .x2 – 6x + 9 = 0 Example 1 .9x2 + 36x - 27 = 0 9 x 2 36x 27 + =0 9 9 9 .x2 + 4x – 3 = 0 a)

.x2 + 73x - 61 = 0

b)

7x2 - 14x - 35 = 0

c)

2x(x + 3) = 4x + 7

d)

2x +

6

-(4) = -4

2 1 = x 4

e) 4x2 + kx + k – 1 = 0


9

-3


2.3. TYPES OF ROOTS QUADRATIC EQUATION

.x =

b

b

2

 4ac



2a .b2 - 4ac

. b2 - 4ac = 0 . Two equal root

. b2 - 4ac > 0 . Two different

2.3.2

. b2 - 4ac < 0 . No real roots

Determine the types of roots for each of the following quadratic equations

Example 1

Example 2

Example 3

a) . x2 – 12x + 27 = 0 . a = 1, b = -12 , c = 27

b) .4 x2 – 12x + 9 = 0 . a = 4, b = -12 , c = 9

c) .2 x2 – 7x + 10 = 0 . a = 2, b = -7 , c = 10

.b2 - 4ac = (-12)2 – 4(1)(27) = 144 – 108 = 36 > 0

.b2 - 4ac = (-12)2 – 4(4)(9) = 144 - 144 = 0

.b2 - 4ac = (-7)2 – 4(2)(10) = 49 – 80 = - 31< 0

Thus, 4x2 – 12x + 9 = 0 Has two equal roots

Thus, 2x2 – 7x + 10 = 0 Has no real roots

Thus, x2 – 12x + 27 = 0 Has two different roots

Exercises 3 1. 2x2 - 8x + 3 = 0

2.

-2y2 + 6x + 3 = 0

3. 3x2 = 7x - 5

4.

4x2 = a(4x - a)



2.4 SOLVING PROBLEMS INVOLVING (.b2 - 4ac ) The discriminant b2 - 4ac of the quadratic equation can be used to a) find an unknown value in an equation Example 1

Example 2

The quadratic equation x2 - 2px + 25 = 0 Has two equal roots. Find the value p

The quadratic equation x2 – 2kx = -(k – 1)2 Has no roots. Find the range of values of k

x2 - 2px + 25 = 0 Thus, a = 1, b = -2p , c = 25 Using .b2 - 4ac = 0 (-2p)2 – 4(1)(25) = 0 4p2 - 100 = 0 4p2 = 100 p2 = 25 p =  25 p = 5

x2 – 2kx = -(k – 2)2 x2 – 2kx + (k – 2)2 = 0 Compare with ax2 + bx + c = 0 Thus .a = 1, b = - 2k , c = (k – 2)2 Using b2 - 4ac < 0 (-2k)2 – 4(k2 -2k + 1)< 0 4k2 - 4k2 + 8k – 4 < 0 8k – 4 < 0 8k < 4 .k < 4

2.4.1

Find the values of m for each of the following quadratic equations which has two equal roots

Exercises 4 1.

mx2 - 4x + 1 = 0

2. x2 – 6x + m = 0

3.

x2 – 2mx + 2m + 3 = 0

4. x2 - 2mx - 4x + 1 = 0



5. x2 + 6x - 9 = m (2x - 3)

2.4.2

6. x2 + 2(x + 2) = m(x2 + 4)

Find the range of values of h for each of the following quadratic equations which roots are different

Exercises 5 1.

x2 - 6x - h = 0

2.

hx2 – 4x – 3 = 0

3.

x2 + 6x + h + 3 = 0

8.

2hx2 + 4x + 1 = 0



5.

2.4.3

x(5 – 2x) = h + 2

6. 2(hx2 – 1) = x(x – 6)

Find the range of values of m for each of the following quadratic equations which has no roots

Exercises 6 1. 2x2 + 2x - m = 0

2.

mx2 + 3x - 3 = 0

3. x2 + 2x + m - 3 = 0

4.

3x2 + 1 = 2(m + 3x)



(h + 1)x2 + 8x + 6 = 0

5.

2.5

6. (m - 3)x2 + 2(1 – m)x = -(m + 1)


SPM 2001/P1 Question 4 1.

Given that -1 and h are roots of the quadratic equation (3x – 1)(x – 2) = p(x – 1), where p is a constant, find the values of h and p


Given that 3 and n are roots the equation (2x + 1)(x – 4) = a(x – 2), where a and n are constants, find the values of a and n.




  and are roots of the quadratic equation px(x – 1) = 3q + x 3 3 If    = 12 and  = 3, find the values of p and q It is given that

SPM 2004/P1 Question 4 4. Form a quadratic equation which has the roots – 5 and Give your answer in the form of ax2 + bx + c = 0 , Where a, b, and c are constans

3 , 4



2.6

ASSESSMENT (30 MINUTES)

1. Form a quadratic equation which has the roots 3 and - 4

2. Given that -3 and 4 are roots of the quadratic equation x2 + ax = b Find the values of a and b

3. The quadratic equation x2 - kx + 2k = 4 has roots 2 and 6 Find the values of k



.4

5.

Find the values of h if the equation x2 = 4hx - 36 has equal roots

The quadratic equation kx2 - 2(3 + k)x = 1 – k has no real roots. Find the range of vales of k

6.

The quadratic equasion x(x – 2m) = - ( 3m + 4) has equal roots, find a) the value of m b) the roots



ANSWERS Exercises 1 a. c. e.

-3, -28, x2 + 3x – 28 = 0

5 2 , , , 3x2 -5x – 2 = 0 3 3 7 2 d. , , 15x2 + 7x – 2 = 0 15 15

b.

5 1 , , 6x2 - 5x + 1 = 0 6 6 21k 18k 2 , , 5x2 - 21kx – 18 k2 = 0 5 5

Exercises 2 a. c. e.

-73 , - 61 7 -1, , 2  k k 1 , 4 4

b. 2, - 5 1 d. ,1 8

Exercises 3 1. 3.

Two different roots No real roots

2. Two different roots 4. Two equal roots

Exercises 4 1. 3.

m=4 m = 3 and -1

5.

m = 3 and 6

2. m = 9 4. m = - 3 and -1 3 1 6. m = and 2 2

Exercises 5 1.

h> - 9

2.

3.

h < 6 9 h< 8

4.

5.

6.

4 3 h < 2 7 h > 4

h >



Exercises 5 1 2 m > 4 5 m> 3

3 4 4. m < -3 1 6. m > 3

1. m < 3. 5.

2.

m