If - - are the roots of 2 x 2-2 x -3-0 and -1- -1 are the roots of Ax 2- B x - C -0- then the value of B - A 2-4 C - A i

Α, β are the roots of 2x^2 2x+ 3 = 0 Difference of roots is, |α β|=√(D_1)|a| ⇒ |α β|=√(4 24)2 ⇒ |α β|=√( 20)2 ...(1) α 1, β 1 are the roots of Ax^2+Bx+C = ...

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Standard XIII

Mathematics

Relation between Roots and Coefficients for Quadratic

If α, β are t...

Question

If $α, β$ are the roots of $2 x^{2} - 2 x + 3 = 0$ and $α - 1, β - 1$ are the roots of $A x^{2} + B x + C = 0,$ then the value of ${(\frac{B}{A})}^{2} - 4 (\frac{C}{A})$ is

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Solution

$α, β$ are the roots of $2 x^{2} - 2 x + 3 = 0$
Difference of roots is,
$| α - β | = \frac{\sqrt{D_{1}}}{| a |}$
$\Rightarrow | α - β | = \frac{\sqrt{4 - 24}}{2}$
$\Rightarrow | α - β | = \frac{\sqrt{- 20}}{2} . . . (1)$

$α - 1, β - 1$ are the roots of $A x^{2} + B x + C = 0$
Difference of roots is,
$| (α - 1) - (β - 1) | = \frac{\sqrt{D_{2}}}{| A |}$
$\Rightarrow | α - β | = \frac{\sqrt{B^{2} - 4 A C}}{| A |} . . . (2)$

From $(1)$ and $(2),$
$\frac{\sqrt{B^{2} - 4 A C}}{| A |} = \frac{\sqrt{- 20}}{2}$
Squaring both sides, we get
$\frac{B^{2} - 4 A C}{A^{2}} = \frac{- 20}{4}$
$\Rightarrow {(\frac{B}{A})}^{2} - 4 (\frac{C}{A}) = - 5$

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If α,β are the roots of 2x2−2x+3=0 and α−1,β−1 are the roots of Ax2+Bx+C=0, then the value of (BA)2−4(CA) is

If $α, β$ are the roots of $2 x^{2} - 2 x + 3 = 0$ and $α - 1, β - 1$ are the roots of $A x^{2} + B x + C = 0,$ then the value of ${(\frac{B}{A})}^{2} - 4 (\frac{C}{A})$ is