To solve the equation [tex]\((x+4)^2 = 25\)[/tex], we will follow these steps:
1. Understand the Equation:
The equation [tex]\((x+4)^2 = 25\)[/tex] represents a quadratic equation in the form of a square of a binomial.
2. Take the Square Root of Both Sides:
To eliminate the square, we will take the square root of both sides of the equation. Remember that taking the square root of both sides of an equation can yield both a positive and a negative solution.
[tex]\[
\sqrt{(x+4)^2} = \pm \sqrt{25}
\][/tex]
3. Simplify Both Sides:
The square root of [tex]\((x+4)^2\)[/tex] is [tex]\(x+4\)[/tex], and the square root of 25 is 5.
[tex]\[
x + 4 = \pm 5
\][/tex]
This yields two separate linear equations:
[tex]\[
x + 4 = 5 \quad \text{and} \quad x + 4 = -5
\][/tex]
4. Solve Each Linear Equation:
- For [tex]\(x + 4 = 5\)[/tex]:
[tex]\[
x + 4 = 5
\][/tex]
Subtract 4 from both sides:
[tex]\[
x = 5 - 4
\][/tex]
[tex]\[
x = 1
\][/tex]
- For [tex]\(x + 4 = -5\)[/tex]:
[tex]\[
x + 4 = -5
\][/tex]
Subtract 4 from both sides:
[tex]\[
x = -5 - 4
\][/tex]
[tex]\[
x = -9
\][/tex]
5. Therefore, the solutions are:
[tex]\[
x = 1 \quad \text{and} \quad x = -9
\][/tex]
After finding the solutions, we can match them with the options provided:
A. [tex]\(x = 9\)[/tex] and [tex]\(x = -1\)[/tex]
B. [tex]\(x = -9\)[/tex] and [tex]\(x = -1\)[/tex]
C. [tex]\(x = -9\)[/tex] and [tex]\(x = 1\)[/tex]
D. [tex]\(x = 9\)[/tex] and [tex]\(x = 1\)[/tex]
The correct answer is:
C. [tex]\(x = -9\)[/tex] and [tex]\(x = 1\)[/tex]
