Question

Solve the equation for all values of x by completing the square. Express your answer in simplest form.
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:
\[
x = -2 + \sqrt{15} \quad \text{and} \quad x = -2 - \sqrt{15}
\]