Question

Solve the equation for all values of x by completing the square. Express your answer in simplest form.
x, squared, plus, 14, x, plus, 45, equals, 0
x
2
+14x+45=
0

Answer 1

To solve the equation \( x^2 + 14x + 45 = 0 \) by completing the square, follow these steps:

1. **Move the constant to the other side of the equation:**

\[
x^2 + 14x = -45
\]

2. **Complete the square for the left side:**

Take the coefficient of \( x \), which is \( 14 \), divide it by \( 2 \), and square it:

\[
\left(\frac{14}{2}\right)^2 = 7^2 = 49
\]

Add \( 49 \) to both sides of the equation:

\[
x^2 + 14x + 49 = -45 + 49
\]

Simplifying the right side:

\[
x^2 + 14x + 49 = 4
\]

3. **Rewrite the left side as a squared term:**

The left side can now be factored:

\[
(x + 7)^2 = 4
\]

4. **Take the square root of both sides:**

\[
x + 7 = \pm 2
\]

5. **Solve for \( x \):**

- If \( x + 7 = 2 \):

\[
x = 2 - 7 = -5
\]

- If \( x + 7 = -2 \):

\[
x = -2 - 7 = -9
\]

6. **Final answers:**

The solutions to the equation \( x^2 + 14x + 45 = 0 \) are:

\[
x = -5 \quad \text{and} \quad x = -9
\]

Thus, the values of \( x \) are \( \boxed{-5} \) and \( \boxed{-9} \).