If $2$ is a root of the equation $x^{2} + b x + 12 = 0$ and the equation $x^{2} + b x + q = 0$ has equal roots, then $q =$

$8$

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$- 8$

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$16$

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$- 16$

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Solution

The correct option is C
$16$

Step 1: Find the value of $b$ .
Consider the given equation $x^{2} + b x + 12 = 0$ .
It is given that $2$ is the root of the equation. So substitute $x = 2$ as:
$\begin{array}{rcl} {(2)}^{2} + 2 b + 12 & = & 0 \\ 4 + 2 b + 12 & = & 0 \\ 2 b + 16 & = & 0 \\ 2 b & = & - 16 \\ b & = & - 8 \end{array}$
Step 2: Find the value of $q$ .
Now, consider the equation $x^{2} + b x + q = 0$ has equal roots.
Therefore the discriminant is equal to zero. Hence:
$\begin{array}{rcl} b^{2} - 4 a c & = & 0 \\ {(- 8)}^{2} - 4 (1) (q) & = & 0 \\ 64 - 4 q & = & 0 \\ - 4 q & = & - 64 \\ q & = & 16 \end{array}$
Hence, the required value is $q = 16$ . Thus, option (C) is correct.

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