Solve the equation for all values of x by completing the square. Express your answer in simplest form.

Question

x, squared, minus, 11, equals, minus, 4, x
x
2
−11=
−4x

Answer 1

To solve the equation \( x^2 - 11 = -4x \) by completing the square, we first rearrange the equation to bring all terms to one side.

1. Start with the given equation: \[ x^2 - 11 + 4x = 0 \]

2. Rearranging gives us: \[ x^2 + 4x - 11 = 0 \]

3. Now, we will complete the square. To complete the square for the quadratic \( x^2 + 4x \), we need to take half of the coefficient of \( x \) (which is 4), square it, and add and subtract it inside the equation.

   Half of 4 is 2 and squaring it gives us: \[ 2^2 = 4 \]

4. Add and subtract 4 in the equation: \[ x^2 + 4x + 4 - 4 - 11 = 0 \implies (x + 2)^2 - 15 = 0 \]

5. Now, isolate the squared term: \[ (x + 2)^2 = 15 \]

6. Take the square root of both sides: \[ x + 2 = \pm \sqrt{15} \]

7. Finally, solve for \( x \): \[ x = -2 + \sqrt{15} \quad \text{and} \quad x = -2 - \sqrt{15} \]

Thus, the solutions are: \[ x = -2 + \sqrt{15} \quad \text{and} \quad x = -2 - \sqrt{15} \]