Question

Solve x^2 + 4x = 27 Correctly to complete the square

Answer 1

To solve the equation \( x^2 + 4x = 27 \) by completing the square, follow these steps:

1. **Move 27 to the left side of the equation**:
\[
x^2 + 4x - 27 = 0
\]

2. **Isolate the \(x\) terms**:
\[
x^2 + 4x = 27
\]

3. **Complete the square**: To do this, take half of the coefficient of \(x\) (which is 4), square it, and add that value to both sides of the equation. Half of 4 is 2, and \(2^2 = 4\).

Add 4 to both sides:
\[
x^2 + 4x + 4 = 27 + 4
\]

This simplifies to:
\[
(x + 2)^2 = 31
\]

4. **Take the square root of both sides**:
\[
x + 2 = \pm \sqrt{31}
\]

5. **Solve for \(x\)**:
\[
x = -2 \pm \sqrt{31}
\]

Thus, the solutions to the equation \( x^2 + 4x = 27 \) are:
\[
x = -2 + \sqrt{31} \quad \text{and} \quad x = -2 - \sqrt{31}
\]