The common root of the quadratic equations $x^{2} - 9 = 0$ and $x^{2} + 6 x + 9 = 0$ is .

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-3

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-2

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Solution

The correct option is C -3
Using the identity $a^{2} - b^{2} = (a - b) (a + b),$ we have
$x^{2} - 9 = (x - 3) (x + 3) .$
Hence, $x = 3, - 3$ are the roots of $x^{2} - 9 = 0.$

Using the identity $(a + b)^{2} = a^{2} + 2 a b + b^{2},$ we can have the following interpretation.
$x^{2} + 6 x + 9 = x^{2} + 2 (x) (3) + (3)^{9} = (x + 3)^{2}$
Thus, the roots of the equation $x^{2} + 6 x + 9 = 0$ are $x = - 3, - 3.$

Hence, the common root of the equations $x^{2} - 9 = 0$ and $x^{2} + 6 x + 9 = 0$ is -3.

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