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: \[ x = -2 + \sqrt{31} \quad \text{and} \quad x = -2 - \sqrt{31} \]