Solve the equation [tex]-2(x-4)^2 + 3 = -69[/tex]. Show every step.

1. Subtract 3 from both sides:
[tex]\[
-2(x-4)^2 + 3 - 3 = -69 - 3
\][/tex]
[tex]\[
-2(x-4)^2 = -72
\][/tex]

2. Divide both sides by -2:
[tex]\[
\frac{-2(x-4)^2}{-2} = \frac{-72}{-2}
\][/tex]
[tex]\[
(x-4)^2 = 36
\][/tex]

3. Take the square root of both sides:
[tex]\[
\sqrt{(x-4)^2} = \pm \sqrt{36}
\][/tex]
[tex]\[
x-4 = \pm 6
\][/tex]

4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x - 4 = 6 \quad \text{or} \quad x - 4 = -6
\][/tex]
[tex]\[
x = 10 \quad \text{or} \quad x = -2
\][/tex]

Therefore, the solutions are:
[tex]\[
x = 10 \quad \text{or} \quad x = -2
\][/tex]



Answer :

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]