Find the roots of the following quadratic equations by factorisation:
(i) x2 – 3x – 10 = 0
(ii) 2x2 + x – 6 = 0
(iii) √2 x2 + 7x + 5√2 = 0
(iv) 2x2 – x + 1/8 = 0
(v) 100x2 – 20x + 1 = 0
(i) x2 – 3x – 10
= x2 - 5x + 2x - 10
= x(x - 5) + 2(x - 5)
= (x - 5)(x + 2)
Roots of this equation are the values for which (x - 5)(x + 2) = 0
∴ x - 5 = 0 or x + 2 = 0
⇒ x = 5 or x = -2
(ii) 2x2 + x – 6
= 2x2 + 4x - 3x - 6
= 2x(x + 2) - 3(x + 2)
= (x + 2)(2x - 3)
Roots of this equation are the values for which (x + 2)(2x - 3) = 0
∴ x + 2 = 0 or 2x - 3 = 0
⇒ x = -2 or x = 3/2
(iii) √2 x2 + 7x + 5√2
= √2 x2 + 5x + 2x + 5√2
= x (√2x + 5) + √2(√2x + 5)= (√2x + 5)(x + √2)
Roots of this equation are the values for which (√2x + 5)(x + √2) = 0
∴ √2x + 5 = 0 or x + √2 = 0
⇒ x = -5/√2 or x = -√2
(iv) 2x2 – x + 1/8
= 1/8 (16x2 - 8x + 1)
= 1/8 (16x2 - 4x -4x + 1)
= 1/8 (4x(4x - 1) -1(4x - 1))
= 1/8(4x - 1)2
Roots of this equation are the values for which (4x - 1)2 = 0
∴ (4x - 1) = 0 or (4x - 1) = 0
⇒ x = 1/4 or x = 1/4
(v) 100x2 – 20x + 1
= 100x2 – 10x - 10x + 1
= 10x(10x - 1) -1(10x - 1)
= (10x - 1)2
Roots of this equation are the values for which (10x - 1)2 = 0
∴ (10x - 1) = 0 or (10x - 1) = 0
⇒ x = 1/10 or x = 1/10
