Question

If ​​​​​​$x^2+px+q=0$ and $x^2+qx+p=0$ have a common root, then ____.

• A. p = q
• B. p + q = -1
• C. p + q = 0
• D. Both (a) and (b)

Answer 1

The correct option is D Both (a) and (b)

Let $\alpha$ be the common root :
$\therefore {\alpha }^2+p\alpha +q=0\left(1\right)$
and
${\alpha }^2+q\alpha +p=0\left(2\right)$

Solving (1) & (2),we get,
$\frac{{\alpha }^2}{p^2-q^2}=\frac{\alpha }{q-p}=\frac{1}{q-p}\left[\text{cross multiplication method of linear pair}\right]$
$\therefore \alpha =\frac{p^2-q^2}{q-p}and\alpha =1$
$\Rightarrow \frac{p^2-q^2}{q-p}=1\Rightarrow p^2-q^2=q-p$
$\Rightarrow (p^2-q^2)+(p-q)=0$
$\Rightarrow (p-q)(p+q+1)=0$
$\Rightarrow p-q=0orp+q+1=0$