Question

2x^2+8x-3=0 by completing the square

Answer 1

To solve the equation \(2x^2 + 8x - 3 = 0\) by completing the square, follow these steps:

1. **Move the constant term to the other side of the equation**:
\[
2x^2 + 8x = 3
\]

2. **Factor out the coefficient of \(x^2\) from the left-hand side (which is 2)**:
\[
2(x^2 + 4x) = 3
\]

3. **Divide both sides by 2**:
\[
x^2 + 4x = \frac{3}{2}
\]

4. **Complete the square**:
Take half of the coefficient of \(x\) (which is 4), square it, and add it to both sides. Half of 4 is 2, and \(2^2 = 4\).
\[
x^2 + 4x + 4 = \frac{3}{2} + 4
\]
\[
x^2 + 4x + 4 = \frac{3}{2} + \frac{8}{2} = \frac{11}{2}
\]

5. **Rewrite the left-hand side as a perfect square**:
\[
(x + 2)^2 = \frac{11}{2}
\]

6. **Take the square root of both sides**:
\[
x + 2 = \pm \sqrt{\frac{11}{2}}
\]

7. **Solve for \(x\)**:
\[
x = -2 \pm \sqrt{\frac{11}{2}}
\]

Thus, the solutions are:
\[
x = -2 + \sqrt{\frac{11}{2}} \quad \text{and} \quad x = -2 - \sqrt{\frac{11}{2}}
\]