Question

If the equations $2x^2+3x+5\lambda =0$ and $x^2+3x+3\lambda =0$ have a common root, then $\lambda$ is equal to

• A. $0,-18$
• B. $-1$
• C. $0,-1$
• D. $2,-1$

Answer 1

The correct option is A $0,-18$

Let $\alpha$ be the common root.
Therefore,
$2(\alpha {)}^2+3\alpha +5\lambda =0$...(i)
$(\alpha {)}^2+3\alpha +3\lambda =0$...(ii)
Subtracting (ii) from (i), we get
${\alpha }^2+2\lambda =0$
$\alpha =\pm \sqrt{-2\lambda }$
Substituting in $2x^2+3x+5\lambda =0$
we get
$2(-2\lambda )\pm 3\left(\sqrt{-2\lambda }\right)+5\lambda =0$

$\lambda =\mp 3\sqrt{-2\lambda }$
Hence, $\lambda =0$ or $\lambda =-18$