(Q4)Find the roots of the follwing quadratic equations by factorisation,

(i)x2 - 3x - 10 = 0  (ii) 2x2 + x - 6 = 0 (iii)√2x2 + 7x + 5√2 = 0 (iv)2x 2 - x +
1
8
 = 0 (v)100x2- 20x + 1 = 0 (vi)x(x + 4) = 12 (vii)3x2- 5x + 2 = 0  (viii)x -
3
x
 = 2(x≠0) (ix)3(x - 4)2- 5(x - 4) = 12

Answer:

(i) Given x2 - 3x - 10 = 0

x2-x + 2x - 10 = 0

x(x - ) + 2(x - ) = 0

(x - )(x + 2) = 0

x - = 0 or x + 2 = 0

x = or x = -2

x = or -2 are the roots of the given quadratic equation.

(ii)Given 2x 2 + x - 6 = 0

2x 2 + 4x - x - 6 = 0

2x(x + 2) - (x + 2) = 0

(x + 2)(2x - ) = 0

(x + 2) or 2x - = 0

x = -2 or 2x =

x = -2 or
2
 are the roots of the quadratic equation.
(iii)Given √2x2 + 7x + 5√2 = 0

√2x2+x + 2x + 5√2 = 0

x(√2x + ) + √(√2x + ) = 0

(√2x + )(x + √)= 0

√2x + = 0 or x + √= 0

√2x = - or x = -√

x =
-
√2
 or - √ are the roots of the given quadratic equation.
(iv)Given 2x2- x +
1
8
= 0
16x2- 8x + 1
8
= 0
16x2- 8x + 1 = 0

16x2-x - 4x + 1 = 0

x(4x - 1)-(4x - 1) = 0

(x - )(4x - 1) - 0

x - = 0

x =

x =
1
,
1
4
 are the roots of given quadratic equation.
(v)Given 100x2- 20x + 1 = 0

100x2 - 10x - 10x + 1 = 0

10x(10x - 1) - 1(10x - 1) = 0

(10x - 1)(10x - 1) = 0

10x - 1 = 0

10x = 1

x =
1
10
,
1
10
are the roots of the given quadratic equation.

(vi)Given x(x + 4) = 12

x2 + 4x = 12

x2 + 4x - 12 = 0

x2+ x - 2x - 12 = 0

x(x + ) - 2(x+) = 0

(x + )(x - 2) = 0

x + = 0 or x - 2 = 0

x = - or 2 are the roots of the given quadratic equation.

(vii)Given 3x2 - 5x + 2 = 0

3x2-x - 2x + 2 = 0

x(x - 1)-(x - 1) = 0

(x - 1)(x - ) = 0

x - 1 = 0 or x- = 0

x = 1 or
are the roots of the quadratic equation.
(viii)Given x-
3
x
 = 2
x2-3
x
 = 2
x2 - 3 = 2x

x2 - 2x - 3 = 0

x2-x + x - 3 = 0

x(x - )+ 1(x -) = 0

(x - )(x + 1) = 0

x - = 0 or x + 1 = 0

x = or -1 are the roots of the given equation.

(ix)Given 3(x - 4)2 - 5(x - 4) = 12

Take (x - 4) = a, then the given quadratic equation reduces to 3a2- 5a = 12

3a2- 5a - 12 = 0

3a2- 9a + a - 12 = 0

3a(a - 3) + (a - 3) = 0

a - 3 = 0 or 3a + = 0

a = 3 or a =
-
3
 but a = x - 4 x - 4 = 3 (or) x - 4 =
-
3
 x = 7 (or) x = 4 -
-
3
=
3
 x = 7 or
3
 are the roots of the quadratic equation.