Question

The quadratic equations $x^2+(a^2-2)x-2a^2=0$ and $x^2-3x+2=0$ have

• A. no common root for all $a\in R$
• B. exactly one common root for all $a\in R$
• C. two common roots for some $a\in R$
• D. none of these

Answer 1

Answer: (B)

exactly one common root for all $a\in R$

Given,
$x^2+(a^2-2)x-2a^2=0$ and $x^2-3x+2=0$
$x^2-3x+2=0$
$\Rightarrow (x-1)(x-2)=0$
$x=1,2$ are the roots of this equation.
$x^2+(a^2-2)x-2a^2=0$
$x^2+a^2-2x-2a^2=0$
$x(x+a^2)-2(x+a^2)=0$
$(x-2)(x+a^2)=0$
$x=2$,$-a^2$
For all $a\in R,-a^2\ne 1$.
So, there will be exactly one common root for all $a\in R$.
Hence, option 'B' is correct.