Question

If the roots of the equation (a² + b²)x² – 2(ac + bd)x + (c² + d²) = 0 are equal, prove that ad = bc.

Answer 1

Given: The quadratic (a² + b²)x² – 2(ac + bd)x + (c² + d²) = 0 has equal roots.

To Prove: ad = bc

Proof [Step-wise]:

1. If a quadratic Ax² + Bx + C = 0 has equal roots then its discriminant Δ = B² – 4AC = 0 (discriminant condition).

2. Here A = a² + b², B = –2(ac + bd), C = c² + d².

So compute Δ: Δ = [–2(ac + bd)]² – 4(a² + b²)(c² + d²).

3. Simplify Δ: Δ = 4(ac + bd)² – 4(a² + b²)(c² + d²).

4. Using Δ = 0, divide by 4: (ac + bd)² = (a² + b²)(c² + d²).

5. Expand both sides: Left: a²c² + 2ac × bd + b²d².

Right: a²c² + a²d² + b²c² + b²d².

6. Subtract the common terms a²c² + b²d² from both sides: 2ac × bd = a²d² + b²c².

7. Rearranging gives: 0 = a²d² – 2ac × bd + b²c² = (ad – bc)².

8. Hence (ad – bc)² = 0, so ad – bc = 0.