Solve the quadratic equation 2x² + ax – a² = 0 for x.

Method 1 (By factorization):

Given quadratic equation, 2x² + ax – a² = 0

⇒ 2x² + 2ax – ax – a² = 0

⇒ 2x (x + a) – a (x + a) = 0

⇒ (x + a) (2x – a) = 0

⇒ x + a = 0 (or) 2x – a = 0

⇒ x = -a (or) x = a/2

Ans. The roots of the given equation are x = –a, a/2.

2^nd method (By Quadratic formula):

Comparing the given equation with ax² + bx + c = 0, we get

a = 2, b = a, c = -a²

Let us calculate b² – 4ac.

⇒ a² – 4(2) (-a²)

⇒ a² + 8a² = 9a² ≥ 0

We know that if b² – 4ac ≥ 0, then the roots of the quadratic equation ax² + bx + c = 0 are given by .

⇒ x = (-a � √9a²) /4

⇒ x = (-a � 3a)/4

⇒ x = (-a + 3a)/4 (or) x = (-a – 3a)/4

⇒ x = 2a/4 (or) x = -4a/4

⇒ x = a/2, x = -a

Ans. The roots of the given equation are x = –a, a/2.