Question

The equations $x^2+x+a=0$ and $x^2+ax+1=0$ have a common real root

• A. For no value of $a$
• B. For exactly one value of $a$
• C. For exactly two values of $a$
• D. For exactly three values of $a$

Answer 1

The correct option is B For exactly one value of $a$

Let $\alpha$ be the common roots
$\therefore {\alpha }^2+\alpha +a=0$ ..... (i)
and

${\alpha }^2+a\alpha +1=0$ ..... (ii)

$\frac{{\alpha }^2}{1-a^2}=\frac{\alpha }{a-1}=\frac{1}{a-1}[a\ne 1]$

Eliminating $\alpha$, we get
$(a-1{)}^2=(1-a^2\left)\right(a-1)$
$\Rightarrow (a-1)=1-a^2$
$\Rightarrow a^2+a-2=0$
$\Rightarrow a^2+2a-a-2=0$
$\Rightarrow a(a+2)-1(a+2)=0$
$\Rightarrow (a+2)(a-1)=0$
$\Rightarrow a=-2[\because a\ne 1]$