Sure, let's solve the given quadratic equation step by step:
The original equation is:
[tex]\[ 4(x + 2)^2 + 13 = 29 \][/tex]
Step 1: Simplify the equation by isolating the quadratic term.
First, subtract 13 from both sides of the equation to remove the constant term on the left-hand side.
[tex]\[
4(x + 2)^2 + 13 - 13 = 29 - 13
\][/tex]
[tex]\[
4(x + 2)^2 = 16
\][/tex]
Step 2: Simplify the coefficient of the quadratic term.
Next, divide both sides of the equation by 4 to further isolate [tex]\((x + 2)^2\)[/tex].
[tex]\[
\frac{4(x + 2)^2}{4} = \frac{16}{4}
\][/tex]
[tex]\[
(x + 2)^2 = 4
\][/tex]
Step 3: Solve for [tex]\(x + 2\)[/tex].
Take the square root of both sides of the equation to eliminate the square. Remember to consider both the positive and negative square roots.
[tex]\[
x + 2 = \pm\sqrt{4}
\][/tex]
[tex]\[
x + 2 = \pm2
\][/tex]
Step 4: Solve for [tex]\(x\)[/tex].
Lastly, solve for [tex]\(x\)[/tex] by subtracting 2 from both sides in each case.
1. For [tex]\(x + 2 = 2\)[/tex]:
[tex]\[
x + 2 = 2
\][/tex]
[tex]\[
x = 2 - 2
\][/tex]
[tex]\[
x = 0
\][/tex]
2. For [tex]\(x + 2 = -2\)[/tex]:
[tex]\[
x + 2 = -2
\][/tex]
[tex]\[
x = -2 - 2
\][/tex]
[tex]\[
x = -4
\][/tex]
Summary:
The values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(4(x + 2)^2 + 13 = 29\)[/tex] are:
[tex]\[
x = 0 \quad \text{and} \quad x = -4
\][/tex]
So, the solutions to the equation are:
[tex]\[
\boxed{0 \text{ and } -4}
\][/tex]
