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Question

# Solve each absolute value equation.$a. \left| x ^ { 2 } - 3 x - 14 \right| = 4 \quad b. x ^ { 2 } = | x | + 6$

Solution

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Since for any $c>0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the given equation, $|x^2-3x-14|=4 ,$ is equivalent to

\begin{align*} x^2-3x-14&=-4 \\\\\text{OR}\\\\ x^2-3x-14&=4 .\end{align*}

Solving each equation results to

\begin{align*} x^2-3x-14+4&=0 \\ x^2-3x-10&=0 \\ (x-5)(x+2)&=0 \\ x&=\{-2,5\} \\\\\text{OR}\\\\ x^2-3x-14&=4 \\ x^2-3x-14-4&=0 \\ x^2-3x-18&=0 \\ (x-6)(x+3)&=0 \\ x&=\{ -3,6 \} .\end{align*}

Hence, $x=\left\{ -3,-2,5,6 \right\} .$

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