Let's solve the given equation [tex]\( x^2 + 5x = 2 \)[/tex] by completing the square:
1. Given:
[tex]\[
x^2 + 5x = 2
\][/tex]
2. We complete the square on the left-hand side:
[tex]\[
x^2 + 5x + \left(\frac{5}{2}\right)^2 = 2 + \left(\frac{5}{2}\right)^2
\][/tex]
3. Simplify:
[tex]\[
x^2 + 5x + \left(\frac{5}{2}\right)^2 = 2 + \left(\frac{5}{2}\right)^2
\][/tex]
[tex]\[
x^2 + 5x + \frac{25}{4} = 2 + \frac{25}{4}
\][/tex]
[tex]\[
x^2 + 5x + \frac{25}{4} = \frac{8}{4} + \frac{25}{4}
\][/tex]
[tex]\[
x^2 + 5x + \frac{25}{4} = \frac{33}{4}
\][/tex]
4. Write the left-hand side as a perfect square:
[tex]\[
\left(x + \frac{5}{2}\right)^2 = \frac{33}{4}
\][/tex]
5. Take the square root of both sides:
[tex]\[
x + \frac{5}{2} = \pm \sqrt{\frac{33}{4}}
\][/tex]
[tex]\[
x + \frac{5}{2} = \pm \frac{\sqrt{33}}{2}
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -\frac{5}{2} \pm \frac{\sqrt{33}}{2}
\][/tex]
7. This gives us two solutions:
[tex]\[
x = -\frac{5}{2} + \frac{\sqrt{33}}{2}
\][/tex]
and
[tex]\[
x = -\frac{5}{2} - \frac{\sqrt{33}}{2}
\][/tex]
Now we need to identify which of the given options match the solutions we found.
Given options:
1. [tex]\(\frac{5}{2} + \frac{\sqrt{33}}{4}\)[/tex]
2. [tex]\(\frac{5}{2} + \frac{\sqrt{33}}{2}\)[/tex]
3. [tex]\(\frac{5}{2} - \frac{\sqrt{33}}{2}\)[/tex]
4. [tex]\(-\frac{5}{2} + \frac{\sqrt{33}}{2}\)[/tex]
5. [tex]\(-\frac{5}{2} - \frac{\sqrt{33}}{2}\)[/tex]
Compare with our solutions:
- [tex]\(\frac{5}{2} + \frac{\sqrt{33}}{2}\)[/tex] does not match our solutions.
- [tex]\(\frac{5}{2} - \frac{\sqrt{33}}{2}\)[/tex] does not match our solutions.
- [tex]\(-\frac{5}{2} + \frac{\sqrt{33}}{2}\)[/tex] matches [tex]\(-\frac{5}{2} + \frac{\sqrt{33}}{2}\)[/tex]
- [tex]\(-\frac{5}{2} - \frac{\sqrt{33}}{2}\)[/tex] matches [tex]\(-\frac{5}{2} - \frac{\sqrt{33}}{2}\)[/tex]
- [tex]\(\frac{5}{2} + \frac{\sqrt{33}}{4}\)[/tex] does not match our solutions.
Hence, the correct solutions are:
- [tex]\(-\frac{5}{2} + \frac{\sqrt{33}}{2}\)[/tex]
- [tex]\(-\frac{5}{2} - \frac{\sqrt{33}}{2}\)[/tex]
