For how many values of $m$ will the equation $(m + 1) x^{2} + 2 (m + 3) x + (2 m + 3) = 0$ have equal roots?

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Solution

The given equation is $(m + 1) x^{2} + 2 (m + 3) x + (2 m + 3) = 0$
Here, $a = (m + 1), b = 2 (m + 3)$ and $c = (2 m + 3)$

For equal roots,
$D i s c r i m i n a n t = 0 \Rightarrow b^{2} - 4 a c = 0$
$\Rightarrow [2 (m + 3)]^{2} - 4 (m + 1) (2 m + 3) = 0$
$\Rightarrow [(m + 3)]^{2} - (m + 1) (2 m + 3) = 0$
$\Rightarrow m^{2} + 6 m + 9 - 2 m^{2} - 5 m - 3 = 0$
$\Rightarrow - m^{2} + m + 6 = 0$
$\Rightarrow m^{2} - m - 6 = 0$

It is a quadratic equation, so m will have two solution
$∴$ for $2$ values of m , the equation has equal roots.

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