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If the two equations $x^2-c x+d=0$ and $x^2-a x+b=0$ have one common root and the second has equal roots, then $2(b+d)=$
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The correct answer is:
$a c$
Let roots of $x^2-c x+d=0$ be $\alpha, \beta$ then roots of $x^2-a x+b=0$ be $\alpha, \alpha$
$\therefore \alpha+\beta=c, \alpha \beta=d, \alpha+\alpha=a, \alpha^2=b$
Hence $2(b+d)=2\left(\alpha^2+\alpha \beta\right)=2 \alpha(\alpha+\beta)=a c$
$\therefore \alpha+\beta=c, \alpha \beta=d, \alpha+\alpha=a, \alpha^2=b$
Hence $2(b+d)=2\left(\alpha^2+\alpha \beta\right)=2 \alpha(\alpha+\beta)=a c$
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