Question

If a and b are distinct real numbers, show that the quadratic equation 2(a² + b²)x² + 2(a + b)x + 1 = 0 has no real roots.

Answer 1

The given equation is 2(a² + b²)x² + 2(a + b)x + 1 = 0

∴ D = [2(a + b)]² – 4 × 2(a² + b²) × 1 

= 4(a² + 2ab + b²) – 8(a² + b²) 

= 4a² + 8ab + 4b² – 8a² – 8b² 

= –4a² + 8ab – 4b² 

= –4(a² – 2ab + b²) 

= –4(a – b)² < 0 

Hence, the given equation has no real roots.