Question

8. Solve x^2+2x-8=0 by completing the square.

Answers

Answer
shorter please
Answer
shorter
Answered by GPT-4o mini
To solve the equation \(x^2 + 2x - 8 = 0\) by completing the square, we can follow these steps:

1. **Move the constant term to the other side**:
\[
x^2 + 2x = 8
\]

2. **Complete the square**:
To complete the square, we take the coefficient of \(x\) (which is \(2\)), divide it by \(2\) (getting \(1\)), and then square it (getting \(1\)).
\[
x^2 + 2x + 1 = 8 + 1
\]
Therefore, we add \(1\) to both sides.

3. **Rewrite the left side as a squared term**:
\[
(x + 1)^2 = 9
\]

4. **Take the square root of both sides**:
\[
x + 1 = \pm 3
\]

5. **Solve for \(x\)**:
First, we consider the positive square root:
\[
x + 1 = 3 \\
x = 3 - 1 = 2
\]
Now for the negative square root:
\[
x + 1 = -3 \\
x = -3 - 1 = -4
\]

6. **Final solution**:
The solutions to the equation \(x^2 + 2x - 8 = 0\) are:
\[
x = 2 \quad \text{and} \quad x = -4
\]
Answered by GPT-4o mini
Sure! Here’s a shorter version to solve \(x^2 + 2x - 8 = 0\) by completing the square:

1. **Move the constant**:
\[
x^2 + 2x = 8
\]

2. **Complete the square**:
\[
x^2 + 2x + 1 = 9 \quad \text{(Add 1 to both sides)}
\]
\[
(x + 1)^2 = 9
\]

3. **Take square roots**:
\[
x + 1 = \pm 3
\]

4. **Solve for \(x\)**:
- \(x + 1 = 3 \implies x = 2\)
- \(x + 1 = -3 \implies x = -4\)

**Final solutions**:
\[
x = 2 \quad \text{and} \quad x = -4
\]
Answered by GPT-4o mini
Sure! Here's an even shorter solution:

1. **Rearrange**: \(x^2 + 2x = 8\)

2. **Complete the square**:
\[
(x + 1)^2 = 9
\]

3. **Solve**:
\[
x + 1 = \pm 3 \implies x = 2 \text{ or } x = -4
\]

**Solutions**: \(x = 2\) and \(x = -4\)

Related Questions