Q15 of 242 Page 8

If the equation has equal roots, prove that .

For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac

If D = 0, roots are equal

⇒ D = 4m²c² – 4(c² – a²)(1 + m²) = 0

⇒ m²c² – c² + a² – c²m² + a²m² = 0

⇒ c² = a²(1 + m²)

Prove that both the roots of the equation are real but they are equal only when .

If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations and has real roots.

Find the values of k for which the quadratic equation has equal roots. Also, find these roots.

Find the values of p for which the quadratic equation has equal roots. Also, find these roots.