Question

Solve the equation for all values of x by completing the square.
x, squared, plus, 36, equals, minus, 14, x
x
2
+36=
−14x

Answer 1

To solve the equation \( x^2 + 36 = -14x \) by completing the square, we will first rearrange the equation to bring all terms to one side:

\[
x^2 + 14x + 36 = 0
\]

Next, we can complete the square for the expression \( x^2 + 14x \). To do this, we need to find the value to add and subtract that makes this a perfect square trinomial.

1. Take the coefficient of \( x \) (which is 14), divide it by 2 to get 7, and then square it to get 49.
2. We add and subtract this value within the equation:

\[
x^2 + 14x + 49 - 49 + 36 = 0
\]

This simplifies to:

\[
(x + 7)^2 - 13 = 0
\]

Next, we can move the constant term to the other side:

\[
(x + 7)^2 = 13
\]

Taking the square root of both sides gives:

\[
x + 7 = \pm \sqrt{13}
\]

Now, we can solve for \( x \) by isolating it:

1. For the positive square root:

\[
x = -7 + \sqrt{13}
\]

2. For the negative square root:

\[
x = -7 - \sqrt{13}
\]

Thus, the solutions to the equation \( x^2 + 36 = -14x \) are:

\[
x = -7 + \sqrt{13} \quad \text{and} \quad x = -7 - \sqrt{13}
\]