To solve the equation [tex]\((x - 21)^2 = 25\)[/tex], follow these steps:
1. Understand the Equation:
The given equation is [tex]\((x - 21)^2 = 25\)[/tex]. This implies that the square of the expression [tex]\((x - 21)\)[/tex] equals 25.
2. Take the Square Root of Both Sides:
To eliminate the square, we take the square root of both sides of the equation:
[tex]\[
\sqrt{(x - 21)^2} = \sqrt{25}
\][/tex]
This simplifies to:
[tex]\[
|x - 21| = 5
\][/tex]
This equation means that the absolute value of [tex]\((x - 21)\)[/tex] is 5.
3. Set Up Two Separate Equations:
When dealing with absolute values, we consider both the positive and negative scenarios:
[tex]\[
x - 21 = 5 \quad \text{or} \quad x - 21 = -5
\][/tex]
4. Solve Each Equation Separately:
- For the first equation [tex]\(x - 21 = 5\)[/tex]:
[tex]\[
x - 21 = 5
\][/tex]
Add 21 to both sides:
[tex]\[
x = 5 + 21
\][/tex]
[tex]\[
x = 26
\][/tex]
- For the second equation [tex]\(x - 21 = -5\)[/tex]:
[tex]\[
x - 21 = -5
\][/tex]
Add 21 to both sides:
[tex]\[
x = -5 + 21
\][/tex]
[tex]\[
x = 16
\][/tex]
5. Conclusion:
The solutions to the equation [tex]\((x - 21)^2 = 25\)[/tex] are:
[tex]\[
x = 26 \quad \text{and} \quad x = 16
\][/tex]
So, the solutions are [tex]\( x = 26 \)[/tex] and [tex]\( x = 16 \)[/tex].
