Question

Rewrite the equation by completing the square.
\[2 x^{2} +7 x +6 = 0\]
\[(x + {}\]

\[)^2 = {}\]

Answer 1

To rewrite the equation \(2 x^{2} + 7 x + 6 = 0\) by completing the square, follow these steps:

1. First, divide the entire equation by 2 to make the coefficient of \(x^2\) equal to 1:

\[
x^2 + \frac{7}{2} x + 3 = 0
\]

2. Rearrange the equation to isolate the constant on the other side:

\[
x^2 + \frac{7}{2} x = -3
\]

3. To complete the square, take half of the coefficient of \(x\) (which is \(\frac{7}{2}\)), square it, and add it to both sides. Half of \(\frac{7}{2}\) is \(\frac{7}{4}\), and squaring it gives \(\left(\frac{7}{4}\right)^2 = \frac{49}{16}\):

\[
x^2 + \frac{7}{2} x + \frac{49}{16} = -3 + \frac{49}{16}
\]

4. Convert -3 to a fraction with a denominator of 16:

\[
-3 = -\frac{48}{16}
\]

5. Now simplify the right side:

\[
-3 + \frac{49}{16} = -\frac{48}{16} + \frac{49}{16} = \frac{1}{16}
\]

6. Now, rewrite the left side as a squared binomial:

\[
\left(x + \frac{7}{4}\right)^2 = \frac{1}{16}
\]

Putting it all together, the completed square form of the equation is:

\[
\left(x + \frac{7}{4}\right)^2 = \frac{1}{16}
\]

So, the final answer is:

\[
(x + \frac{7}{4})^2 = \frac{1}{16}
\]