To solve the quadratic equation [tex]\( 3(x+7)^2 - 32 = -29 \)[/tex], follow these steps:
1. Simplify the equation:
[tex]\[
3(x+7)^2 - 32 = -29
\][/tex]
Add 32 to both sides to isolate the quadratic term:
[tex]\[
3(x+7)^2 - 32 + 32 = -29 + 32
\][/tex]
This simplifies to:
[tex]\[
3(x+7)^2 = 3
\][/tex]
2. Divide both sides by 3:
[tex]\[
\frac{3(x+7)^2}{3} = \frac{3}{3}
\][/tex]
This simplifies to:
[tex]\[
(x+7)^2 = 1
\][/tex]
3. Solve the resulting equation:
Take the square root of both sides:
[tex]\[
x+7 = \pm 1
\][/tex]
This gives us two equations:
[tex]\[
x+7 = 1 \quad \text{or} \quad x+7 = -1
\][/tex]
4. Solve for [tex]\( x \)[/tex] in both cases:
- For [tex]\( x+7 = 1 \)[/tex]:
[tex]\[
x = 1 - 7
\][/tex]
[tex]\[
x = -6
\][/tex]
- For [tex]\( x+7 = -1 \)[/tex]:
[tex]\[
x = -1 - 7
\][/tex]
[tex]\[
x = -8
\][/tex]
Thus, the solutions to the equation [tex]\( 3(x+7)^2 - 32 = -29 \)[/tex] are:
[tex]\[
x = -8 \quad \text{and} \quad x = -6
\][/tex]
