If m and n are root of the equation 1 x - 1 x - 2 = 3 ; where x ≠ 0 and x ≠ 2 ; find m × n .

1 x - 1 x - 2 = 3 ⇒ x - 2 - x x x - 2 = 3 ⇒ x - 2 - x = 3 x ( x - 2 ) ⇒ 3 x 2 - 6 x + 2 = 0 On comparing with the general quadratic equation, a x 2 + b x + c = 0 , we get a = 3 , b = - 6 , c = 2 x = - b ± b 2 - 4 a c 2 a (Quadratic formula) x = 6 ± - 6 2 - 4 ( 3 ) ( 2 ) 2 × 3 x = 3 ± 1 3 It is given that m and n are roots of the equation. So, m = 3 + 1 3 and n = 3 - 1 3 m × n = 3 + 1 3 3 - 1 3 = 2 3

Solve the quadratic equation 8 x 2 - 14 x + 3 = 0 When x ∈ I (integers)

8 x 2 - 14 x + 3 = 0 8 x 2 - 12 x - 2 x + 3 = 0 (Splitting the middle term) 4 x ( 2 x - 3 ) - ( 2 x - 3 ) = 0 ( 2 x - 3 ) ( 4 x - 1 ) = 0 ( 2 x - 3 ) = 0 OR ( 4 x - 1 ) = 0 (Zero Product Rule) x = 3 2 , 1 4 x = 3 2 , 1 4 both are not integers.So the given eqution has no roots.

Solve the quadratic equation 8 x 2 - 14 x + 3 = 0 When x ∈ Q (rational numbers)

8 x 2 - 14 x + 3 = 0 8 x 2 - 12 x - 2 x + 3 = 0 (Splitting the middle term) 4 x ( 2 x - 3 ) - ( 2 x - 3 ) = 0 ( 2 x - 3 ) ( 4 x - 1 ) = 0 ( 2 x - 3 ) = 0 OR ( 4 x - 1 ) = 0 (Zero Product Rule) x = 3 2 , 1 4 x = 3 2 , 1 4 both are rational number.So the roots in the given eqution are x = 3 2 , 1 4