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

Answer 1

The correct option is D

40

Let α be a common root, then

${\alpha }^2+\alpha \alpha +10=0$

and ${\alpha }^2+b\alpha -10=0$

from (i) - (ii),

$(a-b)\alpha +20=0$ ⇒ α = $-\frac{20}{a-b}$

Substituting the value of α in (i), we get

$(-\frac{20}{a-b}{)}^2+a(-\frac{20}{a-b})+10=0$

⇒400 - 20 a(a - b) + $10(a-b{)}^2$ = 0

⇒40 -2$a^2$ + 2ab + $a^2$ + $b^2$ -2ab = 0

⇒$a^2$ - $b^2$ = 40