if ad is not equal to bc the prove the equation (a^2+b^2)x^2+2(ac+bd)x+(c^2+d^2)=0

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Solution

Given equation is
x2 (a2 + b2) + 2x(ac + bd) + (c2 + d2) = 0

The equation will have no real roots only if Discriminant(D) < 0

⇒ b2 – 4ac < 0

⇒ b2 – 4ac

⇒ [2(ac + bd)]2 – 4 (a2 + b2) (c2 + d2)

⇒ 4(ac + bd)2 – 4(a2c2 + a2d2 + b2c2 + b2d2)

⇒ 4(a2c2 + b2d2 + 2abcd) – 4(a2c2 + a2d2 + b2c2 + b2d2)

⇒ 4a2c2 + 4b2d2 + 8abcd – 4a2c2 – 4a2d2 – 4b2c2 – 4b2d2

⇒ – 4a2d2 – 4b2c2 + 8abcd

– 4 [a2d2 + b2c2 – 2abcd]

– 4[(ad – bc)2]

For ad ≠ bc

D = – 4 × [Value of (ad – bc)2]

∴ D always remain negative

So, D < 0

⇒ The given equation has no real roots.

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