If a root of the equations x^2+px+q=0 and x^2 | vedclass

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(C) Let the common root be $y$. Then $y^2+py+q=0$ and $y^2+\alpha y+\beta=0$. Subtracting the two equations: $(y^2+py+q) - (y^2+\alpha y+\beta) = 0$ $

Vedclass · Accepted Answer

$\frac{q-\beta}{\alpha-p}$ or $\frac{p\beta-\alpha q}{q-\beta}$

Vedclass · Answer

(C) Let the common root be $y$. Then $y^2+py+q=0$ and $y^2+\alpha y+\beta=0$. Subtracting the two equations: $(y^2+py+q) - (y^2+\alpha y+\beta) = 0$ $(p-\alpha)y + (q-\beta) = 0$ $y = \frac{\beta-q}{p-\alpha} = \frac{q-\beta}{\alpha-p}$. Alternatively,using the cross-multiplication method for the system: $\frac{y^2}{p\beta-q\alpha} = \frac{y}{q-\beta} = \frac{1}{\alpha-p}$. From the second and third parts,$y = \frac{q-\beta}{\alpha-p}$. From the first and second parts,$y = \frac{p\beta-q\alpha}{q-\beta}$. Since both expressions represent the same common root $y$,the value is $\frac{q-\beta}{\alpha-p}$ or $\frac{p\beta-q\alpha}{q-\beta}$.

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

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