If - - are the roots of a x2-2 b x-c-0 and - - are the roots of A x2-2 B x-C-0- then

Ax^2+bx+c=0 has roots of α and β ∴ Roots= 2b ±√(4b^2 4ac)/2a ∴ Roots= b ±√(b^2 ac)/a ∴α = b + √(b^2 ac)/a,β = b √(b^2 ac)/a Ax^2+2bx+c=0 has roots α +δ , β + ...

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Quantitative Aptitude

Quadratic Equations

If α, β are t...

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If $α, β$ are the roots of $a x^{2} + 2 b x + c = 0$ and $α + δ$ , $β δ$ are the roots of $A x^{2} + 2 B x + C = 0$ , then

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Solution

$a x^{2} + b x + c = 0$ has roots of $α$ and $β$
$∴ R o o t s = \frac{- 2 b \pm \sqrt{4 b^{2} - 4 a c}}{2 a}$
$∴ R o o t s = \frac{- b \pm \sqrt{b^{2} - a c}}{a}$
$∴ α = \frac{- b + \sqrt{b^{2} - a c}}{a}, β = \frac{- b - \sqrt{b^{2} - a c}}{a}$
$A x^{2} + 2 b x + c = 0$ has roots $α + δ, β + δ$
Roots= $\frac{- 2 B 2 \sqrt{4 B^{2} - 4 A C}}{2 A}$
$∴ α + δ = \frac{- B + \sqrt{B^{2} - 4 A C}}{A}, β + δ = \frac{- B - \sqrt{B^{2} - 4 A C}}{A}$
$α - β = \frac{- b + \sqrt{b^{2} - a c} + b + \sqrt{b^{2} - a c}}{a} \Rightarrow α - β = \frac{2 \sqrt{b^{2} - a c}}{a} . . . (1)$
$(α - δ) - (β - δ) = \frac{- B + \sqrt{B^{2} - A C} + B + \sqrt{B^{2} - A C}}{A} \Rightarrow α - β = \frac{2 \sqrt{B^{2} - A C}}{A} . . . (2)$
(1) = (2) $[∵ L H S a r e e q u a l]$
$∴ \frac{2 \sqrt{b^{2} - a c}}{a} = \frac{2 \sqrt{B^{2} - A C}}{A} \Rightarrow \sqrt{\frac{b^{2} - a c}{B^{2} - a c}} = \frac{a}{A} \Rightarrow \frac{b^{2} - a c}{B^{2} - A C} = {(\frac{a}{A})}^{2}$

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