First slide

Theory of equations

Question

If a root of the equations $x^{2} + px + q = 0$ and $x^{2} + ax + β = 0$ is common, then its value will be (where $p \neq α and q \neq β)$

Moderate

A

$\frac{q - β}{α - p}$
B

$\frac{pβ - αq}{q - β}$
C

$\frac{q - β}{α - p} or \frac{pβ - αq}{q - β}$
D

None of these

Solution

Let the common root be y. Then $y^{2} + py + q = 0$ and $y^{2} + α y + β = 0$
on solving by cross multiplication, we have
$\frac{y^{2}}{pβ - qα} = \frac{y}{q - β} = \frac{1}{α - p}$
$∴ y = \frac{q - β}{α - p} and \frac{y^{2}}{y} = y = \frac{pβ - qα}{q - β}$

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