To solve the quadratic equation [tex]\((x - 16)^2 = 256\)[/tex], follow these detailed steps:
1. Understand the equation:
The equation is [tex]\((x - 16)^2 = 256\)[/tex].
2. Take the square root of both sides:
To remove the square, take the square root of both sides. Remember, taking the square root of a squared term introduces both positive and negative roots.
[tex]\[
\sqrt{(x - 16)^2} = \sqrt{256}
\][/tex]
This simplifies to:
[tex]\[
x - 16 = \pm 16
\][/tex]
3. Solve the two resulting equations:
This step generates two separate equations:
[tex]\[
x - 16 = 16 \quad \text{and} \quad x - 16 = -16
\][/tex]
For the first equation:
[tex]\[
x - 16 = 16 \implies x = 16 + 16 = 32
\][/tex]
For the second equation:
[tex]\[
x - 16 = -16 \implies x = 16 - 16 = 0
\][/tex]
4. Combine the solutions:
The solutions are [tex]\( x = 32 \)[/tex] and [tex]\( x = 0 \)[/tex].
Therefore, the correct answer is:
D. [tex]\(x = 32\)[/tex] and [tex]\(x = 0\)[/tex]
