Question

Solve the following quadratic equation for all values of xx in simplest form.
left parenthesis, x, plus, 4, right parenthesis, squared, minus, 38, equals, minus, 22
(x+4)
2
−38=
−22

Answer 1

To solve the quadratic equation \((x + 4)^2 - 38 = -22\), we can start by simplifying the equation.

1. Move \(-22\) to the left-hand side:

\[
(x + 4)^2 - 38 + 22 = 0
\]

This simplifies to:

\[
(x + 4)^2 - 16 = 0
\]

2. Now, we can isolate \((x + 4)^2\):

\[
(x + 4)^2 = 16
\]

3. Next, take the square root of both sides. Remember to consider both the positive and negative square roots:

\[
x + 4 = 4 \quad \text{or} \quad x + 4 = -4
\]

4. Solve for \(x\):

For \(x + 4 = 4\):

\[
x = 4 - 4 = 0
\]

For \(x + 4 = -4\):

\[
x = -4 - 4 = -8
\]

5. Thus, the solutions to the equation are:

\[
\boxed{0} \quad \text{and} \quad \boxed{-8}
\]