If the roots of the equation x2-2bx-c-0 are - and - then b2-c is equal to

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Transformation of Roots: Algebraic Transformation

If the roots ...

Question

If the roots of the equation $x^{2} + 2 b x + c = 0$ are $α$ and $β$ , then $b^{2} - c$ is equal to

$\frac{{(α - β)}^{2}}{4}$
${(α + β)}^{2} - α β$
${(α + β)}^{2} + α β$
$\frac{{(α - β)}^{2}}{2} + α β$
$\frac{{(α + β)}^{2}}{2} + α β$

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Solution

The correct option is A
$\frac{{(α - β)}^{2}}{4}$

Explanation for the correct option:
The given equation is $x^{2} + 2 b x + c = 0$ .
It is known that if $α$ and $β$ are the roots of the equation, then the equation can be given by $x^{2} - (α + β) x + α β = 0$ .
Thus, $α + β = - 2 b$ and $α β = c$ .
$\Rightarrow$ $b = - \frac{α + β}{2}$
Thus, $b^{2} - c = {(- \frac{α + β}{2})}^{2} - α β$ .
$= \frac{{(α + β)}^{2}}{4} - α β = \frac{{(α + β)}^{2} - 4 α β}{4}$
$\Rightarrow$ $b^{2} - c = \frac{{(α - β)}^{2}}{4}$
Therefore, the value of $b^{2} - c$ is equal to $\frac{{(α - β)}^{2}}{4}$ .
Hence, option(A) is the correct option

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If the roots of the equation x2+2bx+c=0 are α and β, then b2-c is equal toα-β24α+β2-αβα+β2+αβα-β22+αβα+β22+αβ

If the roots of the equation $x^{2} + 2 b x + c = 0$ are $α$ and $β$ , then $b^{2} - c$ is equal to

$\frac{{(α - β)}^{2}}{4}$
${(α + β)}^{2} - α β$
${(α + β)}^{2} + α β$
$\frac{{(α - β)}^{2}}{2} + α β$
$\frac{{(α + β)}^{2}}{2} + α β$