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

40.00

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40.0

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Solution

Let α be a common root, then

$\Rightarrow α^{2} + α α + 10 = 0 \dots (i)$

$& α^{2} + b α - 10 = 0 \dots (i i)$

from (i) - (ii), we get:

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

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

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

$\Rightarrow 400 - 20 a (a - b) + 10 (a - b)^{2} = 0 \Rightarrow 40 - 2 a^{2} + 2 a b + a^{2} + b^{2} - 2 a b = 0 \Rightarrow b^{2} - a^{2} = - 40 \Rightarrow a^{2} - b^{2} = 40$

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