If ad bc, then prove that the equation(a2 + b2) x2 + 2 (ac + bd) x + (c2 + d2) = 0 has no real roots.

Question

If ad  bc, then prove that the equation(a2 + b2) x2 + 2 (ac + bd) x + (c2 + d2) = 0 has no real roots.

Philoid · Accepted Answer

We know, that roots of a quadratic equation in general form

Ax² + Bx + C = 0

if and only if D ≥ 0

Where, D = B² - 4AC

In given equation,

A = a² + b²

B = 2(ac + bd)

C = c² + d²

⇒ D = (2ac + 2bd)² - 4(a² + b²)(c² + d²)

⇒ D = 4a²c² + 4b²d² + 8abcd - 4(a²c² + a²d² + b²c² + b²d²)

⇒ D = 4a²c² + 4b²d² + 8abcd - 4a²c² - 4a²d² - 4b²c² - 4b²d²

⇒ D = 8abcd - 4b²c² - 4a²d²

⇒ D = - 4(b²c² + a²d² + 2abcd)

⇒ D = - 4(bc + ad)²

Now, square of any number is always greater than zero,

⇒ (bc + ad)² > 0 [As ad ≠ bc , (bc + ad)² ≠ 0 ]

⇒ - 4(bc + ad)² < 0

⇒ D < 0

⇒ Equation has no real roots

If ad bc, then prove that the equation
(a² + b²) x² + 2 (ac + bd) x + (c² + d²) = 0 has no real roots.