Question

Rewrite the equation by completing the square.
\[x^{2} +7 x +12 = 0\]
\[(x + {}\]

\[)^2 = {}\]

Answer 1

To rewrite the equation \(x^{2} + 7x + 12 = 0\) by completing the square, we follow these steps:

1. Start with the original equation:
\[
x^2 + 7x + 12 = 0
\]

2. Move the constant term to the other side:
\[
x^2 + 7x = -12
\]

3. To complete the square, take the coefficient of \(x\) (which is 7), divide it by 2, and square it:
\[
\left(\frac{7}{2}\right)^2 = \frac{49}{4}
\]

4. Add \(\frac{49}{4}\) to both sides of the equation:
\[
x^2 + 7x + \frac{49}{4} = -12 + \frac{49}{4}
\]

5. Simplify the right side. First, convert \(-12\) into a fraction with a denominator of 4:
\[
-12 = -\frac{48}{4}
\]
So,
\[
-12 + \frac{49}{4} = -\frac{48}{4} + \frac{49}{4} = \frac{1}{4}
\]

6. The left-hand side can be factored since we have completed the square:
\[
(x + \frac{7}{2})^2 = \frac{1}{4}
\]

Thus, the equation by completing the square is:
\[
(x + \frac{7}{2})^2 = \frac{1}{4}
\]

So, you can fill in the blanks as follows:
\[
\begin{align*}
x + \frac{7}{2} \\
\frac{1}{4}
\end{align*}
\]