Question

If $(x+a)$ is the HCF of $x^2+px+q$ and $x^2+lx+m$, then the value of ${}^{\prime }{x}^{\prime }$ is given by-

• A. $\frac{p-l}{q-m}$
• B. $\frac{q-m}{p-l}$
• C. $\frac{l+p}{q+m}$
• D. $\frac{q+m}{l+p}$

Answer 1

The correct option is B $\frac{q-m}{p-l}$

We have,

$(x+a)$ is the HCF of $x^2+px+q$ and $x^2+lx+m$

So,.

Put $x=-a$

Therefore,

$\Rightarrow (-a{)}^2+p(-a)+q=0$

$\Rightarrow a^2-pa+q=0$

$\Rightarrow a^2=pa-q$ $............\left(1\right)$

And

$\Rightarrow (-a{)}^2+l(-a)+m=0$

$\Rightarrow a^2-la+m=0$

$\Rightarrow a^2=la-m$ $............\left(2\right)$

From equations (1) and (2), we get

$la-m=pa-q$

$a(l-p)=m-q$

$a=\frac{m-q}{l-p}$

Therefore, the value of $x$ is

$x=\frac{q-m}{p-l}$