Question
Complete the square to solve the following quadratic equation x squared + 2x - 8 = 0
Answers
GPT-4o mini
To solve the quadratic equation \( x^2 + 2x - 8 = 0 \) by completing the square, follow these steps:
1. **Move the constant term to the other side of the equation:**
\[
x^2 + 2x = 8
\]
2. **Complete the square on the left side.** To do this, take half of the coefficient of \( x \) (which is 2), square it, and add it to both sides. Half of 2 is 1, and \( 1^2 = 1 \).
\[
x^2 + 2x + 1 = 8 + 1
\]
This simplifies to:
\[
x^2 + 2x + 1 = 9
\]
3. **Rewrite the left side as a squared binomial:**
\[
(x + 1)^2 = 9
\]
4. **Take the square root of both sides:**
\[
x + 1 = \pm 3
\]
5. **Solve for \( x \) by isolating it:**
- For the positive case:
\[
x + 1 = 3 \implies x = 3 - 1 = 2
\]
- For the negative case:
\[
x + 1 = -3 \implies x = -3 - 1 = -4
\]
Thus, the solutions to the equation \( x^2 + 2x - 8 = 0 \) are:
\[
x = 2 \quad \text{and} \quad x = -4
\]
1. **Move the constant term to the other side of the equation:**
\[
x^2 + 2x = 8
\]
2. **Complete the square on the left side.** To do this, take half of the coefficient of \( x \) (which is 2), square it, and add it to both sides. Half of 2 is 1, and \( 1^2 = 1 \).
\[
x^2 + 2x + 1 = 8 + 1
\]
This simplifies to:
\[
x^2 + 2x + 1 = 9
\]
3. **Rewrite the left side as a squared binomial:**
\[
(x + 1)^2 = 9
\]
4. **Take the square root of both sides:**
\[
x + 1 = \pm 3
\]
5. **Solve for \( x \) by isolating it:**
- For the positive case:
\[
x + 1 = 3 \implies x = 3 - 1 = 2
\]
- For the negative case:
\[
x + 1 = -3 \implies x = -3 - 1 = -4
\]
Thus, the solutions to the equation \( x^2 + 2x - 8 = 0 \) are:
\[
x = 2 \quad \text{and} \quad x = -4
\]