Question

If x² (a² + b²) + 2x (ac + bd) + c² +d² = 0 has no real roots, then:

Options

• ad ≠ bc

• ad < bc

• ad > bc

• all of these

Answer 1

all of these

Explanation:

If the equation has no real roots then the discriminant of the equation must be less than zero.

⇒ 2² (ac + bd)² − 4 (a² + b²) (c² + d²) < 0

⇒ 4a²c² + 4b²d² + 8acbd < 4a²c² + 4b²d² + 4a²d² + 4b²c²

⇒ 2acbd < a²d² + b²c²

⇒ 2acbd < (ad − bc)² + 2acbd

⇒ (ad − bc)² > 0

⇒ ad ≠ bc and ad < bc or ad > bc