To solve the equation [tex]\((x - 8)^2 = 100\)[/tex], follow these steps:
1. Expand and simplify the equation if necessary: In this case, [tex]\((x - 8)^2\)[/tex] is already simplified on one side of the equation.
2. Take the square root of both sides: To remove the square, we take the square root of both sides of the equation:
[tex]\[
\sqrt{(x - 8)^2} = \sqrt{100}
\][/tex]
3. Simplify the square roots: The square root of [tex]\((x - 8)^2\)[/tex] is the absolute value of [tex]\(x - 8\)[/tex], and the square root of 100 is 10:
[tex]\[
|x - 8| = 10
\][/tex]
4. Create two separate equations: Since the absolute value can be either positive or negative, we split this into two cases:
[tex]\[
x - 8 = 10 \quad \text{or} \quad x - 8 = -10
\][/tex]
5. Solve each equation for [tex]\(x\)[/tex]:
- For [tex]\(x - 8 = 10\)[/tex]:
[tex]\[
x - 8 = 10 \implies x = 10 + 8 \implies x = 18
\][/tex]
- For [tex]\(x - 8 = -10\)[/tex]:
[tex]\[
x - 8 = -10 \implies x = -10 + 8 \implies x = -2
\][/tex]
6. Combine the solutions: The solutions to the equation [tex]\((x - 8)^2 = 100\)[/tex] are [tex]\(x = -2\)[/tex] and [tex]\(x = 18\)[/tex].
Thus, the solutions are:
[tex]\[
x = -2 \quad \text{and} \quad x = 18
\][/tex]
