To solve the quadratic equation [tex]\((x - 16)^2 = 256\)[/tex], follow these steps:
1. Take the square root of both sides:
[tex]\[
\sqrt{(x - 16)^2} = \sqrt{256}
\][/tex]
2. Simplify the equation:
[tex]\[
|x - 16| = 16
\][/tex]
3. Set up the two possible equations based on the absolute value:
[tex]\[
x - 16 = 16 \quad \text{or} \quad x - 16 = -16
\][/tex]
4. Solve each equation separately:
- For [tex]\(x - 16 = 16\)[/tex]:
[tex]\[
x - 16 = 16
\][/tex]
[tex]\[
x = 16 + 16
\][/tex]
[tex]\[
x = 32
\][/tex]
- For [tex]\(x - 16 = -16\)[/tex]:
[tex]\[
x - 16 = -16
\][/tex]
[tex]\[
x = 16 - 16
\][/tex]
[tex]\[
x = 0
\][/tex]
So, the solutions to the equation are [tex]\(x = 32\)[/tex] and [tex]\(x = 0\)[/tex], which corresponds to:
D. [tex]\(x = 32\)[/tex] and [tex]\(x = 0\)[/tex]
