Module 2(a) Add Math - Quadratic Equations

QUADRATIC EQUATIONS

FORM 4

MODULE 2(A) ADDITIONAL MATHEMATICS TOPIC : QUADRATIC EQUATIONS 1.

Find the roots of the equation 2 x 2

+ 5 x = 12 .

[3 marks] [Answer : 2.

3 , −4 ] 2

Given that a root is twice the the other 2 equation 4 x − 18 x + p = 0 , find the value of p .

root root

of

the

quadratic quadratic [4 marks]

[Answer : 8 : 8 ] 3.

Form a quadratic equation which has the the roots roots of -2 and 3. Give your answer in the form of ax 2

+

bx + c = 0 , where a , b and b and c are c are constants. [2 marks]

[Answer : x 4.

2

−

x − 6 = 0 ]

Find the roots of the quadratic equation 2 x 2 to 3 decimal places.

=

5 x + 8 . Give your answer correct [4 marks]

[Answer : 3.609,-1.109 ] 5.

A root of the quadratic equation x 2 root.

+

x + p = 0 is 3. Find the value of the other [3 marks]

[Answer : -4 ] 6.

Given the roots of the quadratic equation px 2

+ 9 x +

4 = 0 is 4 and

1 . Find the 2

value of p . [3 marks] [Answer : 2 ] 7.

Given the equation x 2

− 3x +

k = 0 has the roots of α and β such that

α = −2 . Find the value value of k . β [3 marks] [Answer : -18 ]

8.

Form a quadratic equation which has the roots of in the form of ax 2

+

1 and -2. Give your answer 3

bx + c = 0 , where a , b and b and c are c are constants. [3 marks]

[Answer : 3 x

2

+

5 x − 2 = 0 ]

PROGRAM KECEMERLANGAN AKADEMIK (szk smkpn)

Page 1

QUADRATIC EQUATIONS 9.

If α and β are the roots of the equation 7 x 2 with roots of α + 2 and β + 2 .

+ 5 x + 3 = 0

FORM 4

. Find the equation [3 marks]

[Answer : 7 x 10.

2

− 23 x + 21 =

0 ]

Given that one of the roots of the quadratic equation x 2 negative of the other root, find (a) the value of m , (b) the roots of the equation.

+ (m −

6) x − m = 10 is

[3 marks] [Answer : a) 6 b) 11.

±4

]

Given y = mx is the equation of a tangent to the curve y 2 possible values of m .

=

5 x − 9 . Find the [3 marks]

5 [Answer : ± ] 6 12.

Given that α and β are the roots of the quadratic equation 2 x 2 Form the quadratic equation with the roots of 2α and 2 β .

+

7 x − 15 = 0 [4 marks]

[Answer : x 13.

2

+ 7 x − 30 = 0

]

Given the roots of the equation of 4ax 2 terms of b .

+

bx + 8 = 0 are equal. Express a in [3 marks]

2

[Answer : a = 14.

b ] 128

Diagram 1 shows a right-angled triangle. P x +2 cm Q

3x +2 cm

R

Diagram 1 If the area of

∆PQR

is 16 cm², find the value of x . [4 marks]

[Answer : 2 ] 15.

The quadratic equation x 2 − 2 x + 1 = k ( − x − 2) has two real root and equal roots. Find the possible values of k . [3 marks] [Answer : 0, 12 ]


Page 2


FORM 4

Diagram 2 shows a right-angled triangle STU . S 3x +4 cm 2x -1 cm T

4x cm

U

Diagram 2 Find the value of x . [4 marks] [Answer : 3 ] 17.

The quadratic equation mx 2

− 3mx + 6 =

3x

+

4 has a root of

1 . Find the m

value of m . [4 marks] [Answer : -2 ] 18.

Solve the quadratic equation (5 x − 3)( x + 1) = x (2 x − 5) . Give your answer correct to four significant figures. [3 marks] [Answer : -2.703,0.3699 ]

19.

The quadratic equation ( p + 5)x 2 of p .

= 8 x − 1

has two distinct roots. Find the range [3 marks]

[Answer : p<11 ] 20.

Form the quadratic equation which has the roots of 2 and in the form of ax 2

+

1 . Give your answer 3

bx + c = 0 where a , b and c are constants. [2 marks]

[Answer : 3 x

2

− 7 x + 2 =

0 ]

END OF MODULE


Page 3

QUADRATIC EQUATIONS

FORM 4

ANSWERS MODULE 2(A) ADDITIONAL MATHEMATICS TOPIC : QUADRATIC EQUATIONS 1.

2x 2

+ 5 x − 12 = 0

(2 x − 3)( x + 4) = 0 x 2.

=

3 , x = −4 2

The roots are α and 2α b α + 2α = 3α = − SOR = a

 −18    4 

3α = −  α =

3 2

POR = α (2α ) = 2α 2 2α 2

=

2 = 18

( x + 2)( x − 3) = 0 x2

4.

c a

p 4

3 p = 8    2 3.

=

−

2x 2 x =

x − 6 = 0

− 5 x − 8 = −( −5) ±

0

( −5)2 − 4(2)( −8) 2(2)

5 ± 89 4 = 3.609 or − 1.109 =

5.

x = 3 (3)2 + 3 + p = 0 p = −12 x2

+

x − 12 = 0

( x − 3)( x + 4) = 0 ∴ x = −4

6.

POR = =

c 4  1 = = (4)   a p  2 2


Page 4

QUADRATIC EQUATIONS

7.

SOR = α

+

FORM 4

β = 3 α

=

3 − β

α = −2 β α

= −2β

3 − β

= −2 β

β = −3 α = −2( −3) = 6 k = αβ =

8.

1 ( x − )( x + 2) = 0 3 3x 2

9.

6( −3) = −18

+ 5 x − 2 = 0

SOR = α

+

β

=−

b a

=−

5 7

,

POR

= αβ =

c 3 = a 7

New equation SOR = (α + 2) + ( β + 2) =α +

β + 4

5 23 +4= 7 7 POR = (α + 2)( β + 2) =−

= αβ +

∴ 7x

10.

=

3 7

=

3

2

+

2(α

+

β ) + 4

 5 2−  + 4  7

− 23 x + 21 =

0

Roots α and −α a) SOR = α + ( −α ) = 0 (m − 6) = 0 m = 6 b)

POR

= (α )( −α ) = − m − 10

α 2

=

m + 10

α = ±4 11.

y

=

mx

( mx )2

=

5x − 9

m2 x 2 − 5 x − 9 = 0 Equal roots (touch tangent)


Page 5

QUADRATIC EQUATIONS b2 ( −5)2

−

FORM 4

4ac = 0

2 − 4( m )(9) = 0

25 36 5 m = ± 6

12.

SOR = α

+

β

m 2

=

=−

7 , 2

POR

= αβ = −

15 2

New equation SOR = 2α + 2 β =

2(α

+

β )

 7 2  −  = −7  2 POR = (2α )(2 β ) =

13.

=

4αβ

=

 15  4  −  = −30  2 

x2

+ 7 x − 30 = 0

b2

−

4ac = 0

b2

−

4(4a )(8) = 0

b2

− 128a = 0

b 2 a = 128 14.

1 (3 x + 2)( x + 2) = 16 2 3x 2

+ 8 x − 28 =

0

(3 x + 14)( x − 2) = 0 14 ,2 3 x > 0, x = 2 x = −

15.

x2

− (2 − k ) x + 1 + 2 k = 0

b2

−

4ac = 0 2

[ −(2 − k )] k2

− 4(1)(1 + 2 k ) = 0

− 12k = 0

k ( k − 12) = 0 k

=

0, k = 12


Page 6


(3 x + 4)2 11x 2

= (2 x − 1)

2

+ (4 x )

FORM 4

2

− 28 x − 15 = 0

(11x + 5)( x − 3) = 0 5 ,3 11 x > 0, x = 3 x = −

17.

x =

1 m 2

1 1 1 m   − 3m   + 6 = 3   + 4 m m  m  1 3 −3+6= +4 m m m = −2 18.

5x 2

+

3x 2

+ 7 x − 3 = 0

x = =

2x − 3 = 2 x 2

−7 ±

(7)2

−

− 5x

4(3)( −3)

2(3) −7 ±

85

6 = −2.703 or 0.3699 19.

( p + 5)x 2 b2

−

− 8 x + 1 = 0

4ac > 0

( −8)2

− 4( p + 5)(1) > 0

p < 11 20.

 1 7 SOR = 2 +   = , POR 3 3 7 2 x 2 −   x +   = 0 3 3  3x 2

=

1 2 2   = 3  3

− 7 x + 2 = 0

END OF MODULE


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Module 2(a) Add Math - Quadratic Equations

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