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SPM Practice (Quadratic Equation) 1. Solve the following quadratic equations by

factorisation.

a. 2 5 10 4x x− − = −

b. 23 2 0x x− − =

c. 211 2 12a a= +

d. 2 73 2

x xx

+=

−

[Ans : (a) -1,6 (b) 32

− ,-1 (c) 34

,4 (d) 73

,-1]

2. Solve the following quadratic equations by completing the square.

a. 25 10 3 0x x+ − =

b. − − =22 5 6 0x x

[Ans : (a) -2.265,0.265 (b)-0.866,3.386]

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3. Solve the following quadratic equations by using quadratic formula. Give your answer in four significant figures.

a. ( 1)( 5) 15x x+ − =

b. 2

2

3 2 31

x xx x

+ −=

− −

[Ans : (a) 6.899, -2.899 (b)3.158, -0.158]

4. If the roots of 22 4 1 0x x+ − = are α and β ,

find the equations whose roots are

a.

α 2, β 2

b.

α − β, β − α

[Ans : (a) 24 20 1 0x x− + = (b) 2 6 0x − = ]

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5. Write and simplify the equation whose roots are

double the roots of 23 5 1 0x x− − = , without

solving the given equation.

[Ans : 23 10 4 0x x− − = ]

6. Write and simplify the equation whose roots are

the reciprocals of the roots of

3x2 + 2x −1 = 0,

without solving the given equation.

[Ans : 2 2 3 0x x− − = ]

7. Find the value of p if one root of

x2 + px + 8 = 0 is the square of the other.

[Ans : p=-6]

8. If one root of

2x2 + px + 9 = 0 is twice the

other, find the values of p. [Ans : p=-9,9]

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9. The roots of the equation

6x2 + hx + 1 = 0 are

α and β, where as 3α and 3β are the roots of the

equation

2x2 − x + k = 0. Find the value of h

and k.

[Ans : h= 13

, k=9]

10. Find the range of values of p for which the

equation 22 5 3 0x x p+ + − = has two real

distinct roots.

[Ans : p> 14

− ]

11. The quadratic equation 2 4 1 2 ( 5)x x p x− − = − ,

where p is a constant, has two equal roots. Calculate the possible values of p.

[Ans : p=1,5] 12. Find the range of values of k for which the

equation 2 22 5 6 0x kx k k− + + − = has no real

roots.

[Ans: k> 65

]

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13. Find the range of values of p for which the

equation 25 7 3 6x x p+ − = has no real roots.

[Ans : p< 49260

− ]

14. Show that 2 26 6 2x kx x− − = has no real roots if

14

k > .

15. The quadratic equation

x2 + px + q = 0 has

roots

−2 and 6. Find a. the value of p and of q, b. the range of values of r for which the

equation

x2 + px + q = r has no real roots.

[Ans: a. p=-4,q=-12, b. r<-16]