First slide

Theory of equations

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 ___.

Moderate

A

10
B

20
C

30
D

40

Solution

Let $α$ be a common root, then
$α^{2} + aα + 10 = 0 . . . (i)$
and $α^{2} + bα - 10 = 0 . . . (ii)$
$from (i) - (ii)$
$(a - b) α + 20 = 0 \Rightarrow α = - \frac{20}{a - b}$
Substituting the value of $α$ in (i), we get
${(- \frac{20}{a - b})}^{2} + a (- \frac{20}{a - b}) + 10 = 0$
$\Rightarrow 400 - 20 a (a - b) + 10 {(a - b)}^{2} = 0$
$\Rightarrow 40 - 2 a^{2} + 2 ab + a^{2} + b^{2} - 2 ab = 0 \Rightarrow a^{2} - b^{2} = 40$

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