If the roots of the equation (a 2 b 2 ) x 2 – 2b (a + c) x + (b 2 + c 2 ) = 0 are equal, then

Question

Philoid · Accepted Answer

The roots of the equation are equal so d = b² – 4ac = 0

Here a = (a² + b²), b = – 2b (a + c), c = (b² + c²)

d = (– 2b (a + c))² = 4 (a² + b²) (b² + c²)

⇒ b² (a² + 2ac + c²) = a²b² + a²c² + b⁴ + b²c²

⇒ (ac)² – 2(ac) (b²) + (b²)² = 0

⇒ (ac – b²)² = 0

⇒ (ac – b²) = 0

⇒ a c = b²