Solve the following quadratic equation for all values of xx in simplest form.

Question

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:

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