Question

If the roots of the equation, (a 2 +b 2 )x 2 -2(ac+bd)x+(c 2 +d 2 ) = 0 are equal the
prove that ad = bc.

Answer 1

Answer:

ad = bc

Step-by-step explanation:

Given that :

• The roots of the quadratic equation,                (a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0 are equal.

To prove :

• ad = bc

Proof :

We know that,

The standard form of a quadratic equation is ax² + bx + c = 0. Here,

• a = (a² + b²)
• b = - 2(ac + bd)
• c = (c² + d²)

It is given that the roots are equal, therefore discriminant is zero.

Discriminant = b² - 4ac = 0

⇒ b² - 4ac = 0

⇒ {- 2(ac + bd)}² - 4(a² + b²)(c² + d²) = 0

⇒ 4(ac + bd)² - 4(a² + b²)(c² + d²) = 0

⇒ 4 [ (ac + bd)² - (a² + b²)(c² + d²) ] = 0

⇒ (ac + bd)² - (a² + b²)(c² + d²) = 0

⇒ a²c² + b²d² + 2. ac. bd - a²c² - a²d² - b²c² - b²d² = 0

⇒ - a²d² - b²c² + 2abcd = 0

⇒ - (a²d² + b²c² - 2abcd) = 0

⇒ a²d² + b²c² - 2abcd = 0

⇒ (ad - bc)² = 0

⇒ ad - bc = 0

⇒ ad = bc

           Hence, it is proved.

Answer 2

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• If roots are equal , then ∆ = 0

• Given equation,

• (a²+b²)x² - 2(ac+bd)x + (c²+d²) = 0

Calculating the value of ∆

• ∆ = (2(ac+bd))² - 4(a²+b²)(c²+d²) = 0

• 4(ac)²+4(bd)²+8acbd -4(ac)² -4(ad)² -4(bc)² -4(bd)² = 0

• 8abcd - 4(ad)² -4(bc)² = 0

• -4(ad-bc)² = 0

• ad - bc = 0

• ad = bc

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