Question

If the equations $x^2-kx-21=0$ and $x^2-3kx+35=0$ have a common root $\alpha <0,$ then the value of $k$ is

• A. $-7$
• B. $4$
• C. $7$
• D. $-4$

Answer 1

The correct option is D $-4$

Let $\alpha$ be the common root.
Then, ${\alpha }^2-k\alpha -21=0\cdots \left(1\right)$
and ${\alpha }^2-3k\alpha +35=0\cdots \left(2\right)$
Subtracting $\left(1\right)$ and $\left(2\right),$ we get
$2k\alpha -56=0$
$\Rightarrow k\alpha =28\cdots \left(3\right)$
Putting $k\alpha =28$ in $\left(1\right),$
${\alpha }^2-28-21=0$
$\Rightarrow \alpha =-7(\because \alpha <0)$
From $\left(3\right),$ we have
$k\alpha =28\Rightarrow k=-4$