A quadratic equation ax2-bx-c-0-a0 have the roots - and - Find the transformed equation with roots - and -

Given: nbsp; ax^2+bx+c=0,a0 To find: Transformed equation with roots nbsp; α and β. nbsp; When roots are nbsp; α and β , then nbsp; Sum of roots = ...

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Standard XII

Mathematics

Transformation of Roots: Algebraic Transformation

A quadratic e...

Question

A quadratic equation $a x^{2} + b x + c = 0, a \neq 0$ have the roots $α$ and $β$ . Find the transformed equation with roots $- α$ and $- β .$

a x^{2} + b x + c = 0

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a x^{2} - b x + c = 0

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None of the above

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b x^{2} - a x + c = 0

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Solution

The correct option is B $a x^{2} - b x + c = 0$
Given: $a x^{2} + b x + c = 0, a \neq 0$
To find: Transformed equation with roots $- α$ and $- β .$

When roots are $- α$ and $- β$ , then

Sum of roots $= - α - β = - \frac{b}{a}$

$= - (α + β) = - \frac{b}{a}$

Product of roots $= (- α) . (- β) = \frac{c}{a}$

$= α . β = \frac{c}{a}$

$∴$ Transformed quadratic equation,
$x^{2} - {- (α + β)} x + α . β = 0$

$x^{2} + (- \frac{b}{a}) x + (\frac{c}{a}) = 0$

$a x^{2} - b x + c = 0$

Hence, Transformed equation with roots $- α$ and $- β$ is $a x^{2} - b x + c = 0$

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A quadratic equation ax2+bx+c=0,a≠0 have the roots α and β. Find the transformed equation with roots −α and −β.

A quadratic equation $a x^{2} + b x + c = 0, a \neq 0$ have the roots $α$ and $β$ . Find the transformed equation with roots $- α$ and $- β .$