Sure, I'd be happy to help you solve the equation [tex]\(-2(x-4)^2 + 3 = -69\)[/tex] step-by-step.
### Step-by-Step Solution
1. Start with the given equation:
[tex]\[
-2(x - 4)^2 + 3 = -69
\][/tex]
2. Subtract 3 from both sides to isolate the quadratic term:
[tex]\[
-2(x - 4)^2 + 3 - 3 = -69 - 3
\][/tex]
Simplifying, we get:
[tex]\[
-2(x - 4)^2 = -72
\][/tex]
3. Divide both sides by -2 to make the coefficient of [tex]\((x - 4)^2\)[/tex] equal to 1:
[tex]\[
\frac{-2(x - 4)^2}{-2} = \frac{-72}{-2}
\][/tex]
Simplifying, we get:
[tex]\[
(x - 4)^2 = 36
\][/tex]
4. Take the square root of both sides to solve for [tex]\(x - 4\)[/tex]:
[tex]\[
x - 4 = \pm\sqrt{36}
\][/tex]
Since [tex]\(\sqrt{36} = 6\)[/tex], we have two possible solutions:
[tex]\[
x - 4 = 6 \quad \text{or} \quad x - 4 = -6
\][/tex]
5. Solve each equation for [tex]\(x\)[/tex]:
- For [tex]\(x - 4 = 6\)[/tex]:
[tex]\[
x - 4 = 6 \quad \Rightarrow \quad x = 6 + 4 \quad \Rightarrow \quad x = 10
\][/tex]
- For [tex]\(x - 4 = -6\)[/tex]:
[tex]\[
x - 4 = -6 \quad \Rightarrow \quad x = -6 + 4 \quad \Rightarrow \quad x = -2
\][/tex]
### Final Solutions
Thus, the solutions to the equation [tex]\(-2(x-4)^2 + 3 = -69\)[/tex] are:
[tex]\[
x = 10 \quad \text{and} \quad x = -2
\][/tex]
