Question

If $x^2+ax+10=0$ and $x^2+bx-10=0$ have a common root, then $a^2-b^2$ is equal to:

• A. 10
• B. 20
• C. 30
• D. 40
• E. 50

Answer 1

Answer: (D)

Let $\alpha$ be the common root for both the equations $x^2+ax+10=0$ and $x^2+bx-10=0$ $\therefore$ ${\alpha }^2+a\alpha +10=0$ ...(i) and ${\alpha }^2+b\alpha -10=0$ ...(ii) From Eqs. (i) and (ii), we get $(a-b)\alpha +10+10=0$ $\Rightarrow$ $(a-b)\alpha =-20$ $\Rightarrow$ $\alpha =\frac{-20}{a-b}$ $\because$ $\alpha$ is also the root of $x^2+bx-10=0$ $\therefore$ $\frac{400}{{(a-b)}^2}+b\left(\frac{-20}{a-b}\right)-10=0$ $40-2b(a-b)={(a-b)}^2$ $\Rightarrow$ $a^2+b^2-2b^2=40$ $\Rightarrow$ $a^2-b^2=40$