We are tasked with solving the equation:

$|x^2 + 8x| = 16$

Step 1: Break into two cases based on the absolute value

The absolute value equation means that either:

1. $x^2 + 8x = 16$
2. $x^2 + 8x = -16$

Case 1: $x^2 + 8x = 16$

Rearrange the equation:

$x^2 + 8x - 16 = 0$

Now, solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 1$, $b = 8$, and $c = -16$. Substituting these values into the quadratic formula:

$x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-16)}}{2(1)}$ $x = \frac{-8 \pm \sqrt{64 + 64}}{2}$ $x = \frac{-8 \pm \sqrt{128}}{2}$ $x = \frac{-8 \pm 8\sqrt{2}}{2}$ $x = -4 \pm 4\sqrt{2}$

Thus, the two solutions for this case are:

$x = -4 + 4\sqrt{2}, \quad x = -4 - 4\sqrt{2}$

Case 2: $x^2 + 8x = -16$

Rearrange the equation:

$x^2 + 8x + 16 = 0$

This is a perfect square trinomial:

$(x + 4)^2 = 0$

Thus, the only solution is:

$x = -4$

Final Solutions:

The solutions to the equation are:

$x = -4 + 4\sqrt{2}, \quad x = -4 - 4\sqrt{2}, \quad x = -4$

Would you like more details or have any questions?

Related questions:

1. How do we derive the quadratic formula?
2. Can we factor quadratics instead of using the formula?
3. What are perfect square trinomials?
4. How does absolute value affect the equation?
5. What are the steps to simplify square roots?

Tip: Always check your solutions by plugging them back into the original equation to verify their validity!