To solve the equation [tex]\(x^2 + 2x - 9 = 15\)[/tex] by completing the square, follow these steps:
1. Isolate the constant term:
Move all terms to one side of the equation to isolate the constant.
[tex]\[
x^2 + 2x - 9 - 15 = 0
\][/tex]
Simplify this to:
[tex]\[
x^2 + 2x - 24 = 0
\][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[
x^2 + 2x = 24
\][/tex]
3. Complete the square:
To complete the square, you need to add and subtract the same value to/from the left side of the equation. Here, focus on the [tex]\(x^2 + 2x\)[/tex] part.
Take half of the coefficient of [tex]\(x\)[/tex], square it, and add it inside the equation:
- Coefficient of [tex]\(x\)[/tex] is 2
- Half of it: [tex]\(2/2 = 1\)[/tex]
- Squaring it: [tex]\(1^2 = 1\)[/tex]
Add and subtract 1 to/from the left side:
[tex]\[
x^2 + 2x + 1 - 1 = 24
\][/tex]
This simplifies to:
[tex]\[
(x + 1)^2 - 1 = 24
\][/tex]
4. Simplify the equation:
Move the -1 to the right side:
[tex]\[
(x + 1)^2 = 25
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Take the square root of both sides:
[tex]\[
x + 1 = \pm 5
\][/tex]
Solve for [tex]\(x\)[/tex] in both cases:
- [tex]\(x + 1 = 5\)[/tex]
[tex]\[
x = 5 - 1
\][/tex]
[tex]\[
x = 4
\][/tex]
- [tex]\(x + 1 = -5\)[/tex]
[tex]\[
x = -5 - 1
\][/tex]
[tex]\[
x = -6
\][/tex]
So, the solutions to the equation are [tex]\(x = 4\)[/tex] and [tex]\(x = -6\)[/tex]. Thus, the correct answer is:
C. [tex]\(x = -6\)[/tex]; [tex]\(x = 4\)[/tex]
