Question
Complete the square to solve the following quadratic equation: x^2+2x−8=0
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Responses
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Responses
Answers
GPT-4o mini
To solve the quadratic equation \( x^2 + 2x - 8 = 0 \) by completing the square, follow these steps:
1. Start with the original equation:
\[
x^2 + 2x - 8 = 0
\]
2. Move the constant term to the other side of the equation:
\[
x^2 + 2x = 8
\]
3. To complete the square, take the coefficient of \( x \) (which is 2), divide it by 2, and square it:
\[
\left(\frac{2}{2}\right)^2 = 1
\]
4. Add this square to both sides of the equation:
\[
x^2 + 2x + 1 = 8 + 1
\]
5. This simplifies to:
\[
(x + 1)^2 = 9
\]
6. Now, take the square root of both sides. Remember to consider both the positive and negative roots:
\[
x + 1 = 3 \quad \text{or} \quad x + 1 = -3
\]
7. Solve for \( x \) in each case:
- For \( x + 1 = 3 \):
\[
x = 3 - 1 = 2
\]
- For \( x + 1 = -3 \):
\[
x = -3 - 1 = -4
\]
8. The solutions to the equation are:
\[
x = 2 \quad \text{and} \quad x = -4
\]
Thus, the final answers are \( x = 2 \) and \( x = -4 \).
1. Start with the original equation:
\[
x^2 + 2x - 8 = 0
\]
2. Move the constant term to the other side of the equation:
\[
x^2 + 2x = 8
\]
3. To complete the square, take the coefficient of \( x \) (which is 2), divide it by 2, and square it:
\[
\left(\frac{2}{2}\right)^2 = 1
\]
4. Add this square to both sides of the equation:
\[
x^2 + 2x + 1 = 8 + 1
\]
5. This simplifies to:
\[
(x + 1)^2 = 9
\]
6. Now, take the square root of both sides. Remember to consider both the positive and negative roots:
\[
x + 1 = 3 \quad \text{or} \quad x + 1 = -3
\]
7. Solve for \( x \) in each case:
- For \( x + 1 = 3 \):
\[
x = 3 - 1 = 2
\]
- For \( x + 1 = -3 \):
\[
x = -3 - 1 = -4
\]
8. The solutions to the equation are:
\[
x = 2 \quad \text{and} \quad x = -4
\]
Thus, the final answers are \( x = 2 \) and \( x = -4 \).