(i) x^{2} – 3x – 10 = 0

(ii) 2x^{2} + x – 6 = 0

(iii) √2x^{2} + 7x + 5√2 = 0

(iv) 2x^{2} – x + 1/8 = 0

(v) 100x^{2} – 20x + 1 = 0

**Solution:**

(i) Given, *x*^{2} – 3*x* – 10 = 0

Taking L.H.S.,

*⇒ x*^{2} – 5*x* + 2*x* – 10

*⇒ x*(*x *– 5) + 2(*x* – 5)

*⇒ *(*x* – 5)(*x* + 2)

The roots of this equation,* x*^{2} – 3*x* – 10 = 0 are the values of x for which (*x* – 5)(*x* + 2) = 0

Therefore, *x* – 5 = 0 or *x* + 2 = 0

*⇒ x* = 5 or *x* = -2

(ii) Given, 2*x*^{2} + *x* – 6 = 0

Taking L.H.S.,

*⇒ *2*x*^{2} + 4*x* – 3*x* – 6

*⇒ *2*x*(*x* + 2) – 3(*x* + 2)

*⇒ *(*x* + 2)(2*x* – 3)

The roots of this equation, 2*x*^{2} + *x* – 6=0 are the values of x for which (*x* + 2)(2*x* – 3) = 0

Therefore, *x* + 2 = 0 or 2*x* – 3 = 0

*⇒ x* = -2 or *x* = 3/2

(iii) √2 *x*^{2} + 7*x* + 5√2 = 0

Taking L.H.S.,

*⇒ *√2 *x*^{2 }+ 5*x* + 2*x* + 5√2

*⇒ x* (√2*x* + 5) + √2(√2*x* + 5)= (√2*x* + 5)(*x *+ √2)

The roots of this equation, √2 *x*^{2} + 7*x* + 5√2 = 0 are the values of x for which (√2*x* + 5)(*x *+ √2) = 0

Therefore, √2*x* + 5 = 0 or *x* + √2 = 0

*⇒ x* = -5/√2 or *x* = -√2

(iv) 2*x*^{2} – *x* +1/8 = 0

Taking L.H.S.,

= 1/8 (16*x*^{2 }– 8*x* + 1)

= 1/8 (16*x*^{2 }– 4*x* -4*x* + 1)

= 1/8 (4*x*(4*x* – 1) -1(4*x* – 1))

= 1/8 (4*x *– 1)^{2}

The roots of this equation, 2*x*^{2} – *x* + 1/8 = 0, are the values of x for which (4*x *– 1)^{2}= 0

Therefore, (4*x* – 1) = 0 or (4*x* – 1) = 0

⇒ *x* = 1/4 or *x* = 1/4

(v) Given, 100x^{2} – 20x + 1=0

Taking L.H.S.,

= 100x^{2} – 10x – 10x + 1

= 10x(10x – 1) -1(10x – 1)

= (10x – 1)^{2}

The roots of this equation, 100x^{2} – 20x + 1=0, are the values of x for which (10x – 1)^{2}= 0

∴ (10x – 1) = 0 or (10x – 1) = 0

⇒ x = 1/10 or x = 1/10