Question

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

Answer 1

Let D be the discriminant of the equation  (a² + b²)x² + 2x(ac + bd) + (c² + d²) = 0

Then,

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

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

⇒ D = 4[a²c² + b²d² + 2ac × bd - a²c² - a²d² - b²c² - b²d²]

⇒ D = 4[2ac × bd - a²d² - b²c²]

⇒ D = - 4[a²d² + b²c² - 2ad × bc]

⇒ D = - 4(ad - bc)²

It is given that ad ≠ bc

⇒ ad - bc ≠ 0

⇒ (ad - bc)² > 0

⇒ - 4(ad - bc)² < 0

⇒ D < 0

Hence, the given equation has no real roots.

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