If the roots of the equation (a^2+b^2)x^2-2(ac+bd)x+(c^2+d^2)=0 are equal then prove that ad=bc
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Step-by-step explanation:
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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)
a = (a² + b²) b = - 2(ac + bd)
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.