Given the roots of the equations ax² + 2bx + c = 0 are real.

Comparing with standard quadratic equation Ax² + Bx + C = 0

A = a B = 2b C = c

Discriminant D₁ = B² – 4AC ≥ 0

= (2b)² – 4.a.c ≥ 0

= 4(b² –ac) ≥ 0

= (b² –ac) ≥ 0 – – – – – (1)

For the equation

Discriminant D₂ = b² – 4ac ≥ 0

=

= 4(ac – b²) ≥0

= – 4(b²–ac) ≥0

= (b² –ac) ≥0 – – – – – (2)

The roots of the are simultaneously real if (1) and (2) are true together

b² –ac = 0

b² = ac

Hence proved.