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Complete the square to solve the equation below.

[tex]\[x^2 + 2x - 9 = 15\][/tex]

A. [tex]\(x = 5; x = -4\)[/tex]

B. [tex]\(x = -1; x = 3\)[/tex]

C. [tex]\(x = -6; x = 4\)[/tex]

D. [tex]\(x = -1; x = 4\)[/tex]


Sagot :

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]