If one root of the equations x2-px-q-0 and x2-x-0 is common- then its value will be where p - - and q - -

Let the common root be y, then y^2 + py + q =0 and y^2 + nbsp;αy+β=0 On solving by cross multiplication, we have y^2pβ qα = yq β=1α p ∴ y=q βα p and y^2y = ...

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Standard XIII

Mathematics

Quadratic Equations with Both Roots Common

If one root o...

Question

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

$\frac{q - β}{α - p}$

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$\frac{p β - α q}{q - β}$

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None of these

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$\frac{q - β}{α - p} or \frac{p β - α q}{q - β}$

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Solution

The correct option is D
$\frac{q - β}{α - p} or \frac{p β - α q}{q - β}$

Let the common root be y, then $y^{2} + p y + q = 0$ and $y^{2} + α y + β = 0$

On solving by cross multiplication, we have

$\frac{y^{2}}{p β - q α} = \frac{y}{q - β} = \frac{1}{α - p}$

∴ $y = \frac{q - β}{α - p}$ and $\frac{y^{2}}{y} = y = \frac{p β - q α}{q - β}$

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If one root of the equations x2+px+q=0 and x2+αx+β=0 is common, then its value will be ( where p ≠ α and q ≠ β )

The correct option is D q−βα−p or pβ−αqq−β Let the common root be y, then y2+py+q=0 and y2+αy+β=0 On solving by cross multiplication, we have y2pβ−qα=yq−β=1α−p ∴ y=q−βα−p and y2y=y=pβ−qαq−β

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