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

Answer: (B)

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
$(a-1{)}^2=\left(1-a^2\right)(a-1)$
$\Rightarrow a-1)=1-a^2$
$\Rightarrow a^2+a-2=0$
$\Rightarrow a^2+2a-a-2=0$
$\Rightarrow a(a+2)-1(a+2)=0$
$\Rightarrow (a+2)(a-1)=0$
$\Rightarrow a=-2.[\because a\ne 1]$