Question

The equations $x^2+ax+b=0$ and $x^2+cx+d=0$ have a common root. If the roots of $x^2+ax+b=0$ are equal, then which one of the following options is/are always true?

• A. $a^2=4b$
• B. $c^2=4d$
• C. $2(b+d)=ac$
• D. $2(a+c)=bd$

Answer 1

Answer: (A), (C)

$2(b+d)=ac$

Let $\alpha$ be the common root of $x^2+ax+b=0$ and $x^2+cx+d=0$
Since, roots of $x^2+ax+b=0$ are equal,
$\Rightarrow \alpha +\alpha =-a,{\alpha }^2=b$
$\Rightarrow \alpha =-\frac{a}{2},{\alpha }^2=b$
$\Rightarrow a^2=4b$

Let $\beta$ be the other root of $x^2+cx+d=0$
$\Rightarrow \alpha +\beta =-c,\alpha \beta =d$
$\Rightarrow \alpha +\frac{d}{\alpha }=-c$
$\Rightarrow \frac{{\alpha }^2+d}{\alpha }=-c$
$\Rightarrow b+d=-\alpha c$
$\Rightarrow b+d=\frac{a}{2}c$
$\Rightarrow 2(b+d)=ac$