Question

If $x^2+bx-a=0$ and $x^2-ax+b=0$ have only one common root, then

• A. $a+b=0$
• B. $a=b$
• C. $a-b=1$
• D. $a+b=1$

Answer 1

The correct option is C $a-b=1$

Given equations $x^2+bx-a=0$ and $x^2-ax+b=0$

Let the common root be $\alpha$, then
${\alpha }^2+b\alpha -a=0{\alpha }^2-a\alpha +b=0$
On subtracting, we get
$\alpha (b+a)-(a+b)=0$
$\Rightarrow (a+b)(\alpha -1)=0$
$\Rightarrow a+b=0\text{or}\alpha =1$

When $a+b=0\Rightarrow a=-b$, then both equation becomes
$x^2+bx+b=0$ and $x^2+bx+b=0$
They are same equation, so both roots is same, so rejecting $a+b=0$

When $\alpha =1$, we get
$1+b-a=0\Rightarrow a-b=1$