Question

If the equation $x^2-px+q=0$ and $x^2-ax+b=0$ have a common root and the roots of the second equation are equal, then which one of the following is correct?

Answer 1

f $\alpha \&\beta$ are the roots of

$\mathrm{\$p}{x}^2+qx+r=\mathrm{0\$}$

then

$\mathrm{\$\alpha }\beta =-\frac{q}{\mathrm{p\$}}$

$\alpha \beta =\frac{r}{p}$

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$x^2+ax+b=0---\left(1\right)$

$x^2+cx+d=0---\left(2\right)$

let the common root be $\alpha$

for eqn$\left(1\right)$

$\alpha +\alpha =-a$

$\Rightarrow \alpha =-\frac{a}{2}$

$\text{\&}{\alpha }^2=b$

for the eqn$\left(2\right)$ let the second root be$\beta$

then

$\alpha +\beta =-c$

$\alpha \beta =d$

$\Rightarrow \beta =\frac{d}{\alpha }$

$\therefore \alpha +\frac{d}{\alpha }=-c$

${\alpha }^2+d=\alpha (-c)$

$b+d=(-\frac{a}{2})(-c)$

$\therefore 2(b+d)=ac\text{as reqd.}$