Solve the equation by factorisation: x 2 - 2 + 1 x + 2 = 0

The given equation is: x 2 - 2 + 1 x + 2 = 0 ⇒ x 2 - 2 x - x + 2 = 0 ⇒ x x - 2 - 1 x - 2 = 0 ⇒ x - 2 x - 1 = 0 ⇒ x - 2 = 0 or x - 1 = 0 ⇒ x = 2 or x = 1 Hence, x = 2 and x = 1 are the roots of the given equation.

Solve the equation by factorisation: x - 1 2 2 = 4

The given equation is: x - 1 2 2 = 4 Using ( a - b ) 2 = a 2 + b 2 - 2 a b to expand x - 1 2 2 , we get ⇒ x 2 - 2 x 1 2 + 1 2 2 = 4 ⇒ x 2 - x + 1 4 = 4 ⇒ x 2 - x = 4 - 1 4 ⇒ x 2 - x = 16 - 1 4 ⇒ x 2 - x = 15 4 ⇒ 4 x 2 - x = 15 ⇒ 4 x 2 - 4 x - 15 = 0 Factorising above quadratic equation by splitting the middle term, we get, ⇒ 4 x 2 - 10 x + 6 x - 15 = 0 ⇒ 2 x 2 x - 5 + 3 2 x - 5 = 0 ⇒ 2 x + 3 2 x - 5 = 0 ⇒ 2 x + 3 = 0 or 2 x - 5 = 0 ⇒ 2 x = - 3 or 2 x = 5 ⇒ x = - 3 2 or x = 5 2 Hence, x = - 3 2 and x = 5 2 are the roots of the given equation.