To solve the quadratic equation \(x^2 + 5x = 2\), we will complete the square as shown in the steps provided. Let's go through the process in detail.
Step 1: Start with the given equation:
[tex]\[x^2 + 5x = 2\][/tex]
Step 2: Add \(\left(\frac{5}{2}\right)^2\) to both sides to complete the square:
[tex]\[x^2 + 5x + \left(\frac{5}{2}\right)^2 = 2 + \left(\frac{5}{2}\right)^2\][/tex]
Since \((5/2)^2 = 25/4\), we have:
[tex]\[x^2 + 5x + \frac{25}{4} = 2 + \frac{25}{4}\][/tex]
Step 3: Simplify the right-hand side:
[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]
Step 4: Write the left-hand side as a perfect square:
[tex]\[\left(x + \frac{5}{2}\right)^2 = \frac{33}{4}\][/tex]
Step 5: Solve for \(x\) by taking the square root of both sides:
[tex]\[x + \frac{5}{2} = \pm \frac{\sqrt{33}}{2}\][/tex]
This gives us two equations:
[tex]\[x + \frac{5}{2} = \frac{\sqrt{33}}{2}\][/tex]
[tex]\[x + \frac{5}{2} = -\frac{\sqrt{33}}{2}\][/tex]
Step 6: Solve each equation for \(x\):
For the first equation:
[tex]\[x = \frac{\sqrt{33}}{2} - \frac{5}{2} = \frac{\sqrt{33} - 5}{2}\][/tex]
For the second equation:
[tex]\[x = -\frac{\sqrt{33}}{2} - \frac{5}{2} = -\frac{\sqrt{33} + 5}{2}\][/tex]
Simplify these results to get:
[tex]\[x = -\frac{5}{2} + \frac{\sqrt{33}}{2}\][/tex]
[tex]\[x = -\frac{5}{2} - \frac{\sqrt{33}}{2}\][/tex]
So, the solutions to the equation \(x^2 + 5x = 2\) are:
[tex]\[x = -\frac{5}{2} + \frac{\sqrt{33}}{2}\][/tex]
[tex]\[x = -\frac{5}{2} - \frac{\sqrt{33}}{2}\][/tex]
Now, let's identify which of the given options are equivalent to these solutions.
Given solutions to check:
1. \(\frac{5}{2} + \frac{\sqrt{33}}{4}\)
2. \(\frac{5}{2} + \frac{\sqrt{33}}{2}\)
3. \(\frac{5}{2} - \frac{\sqrt{33}}{2}\)
4. \(-\frac{5}{2} - \frac{\sqrt{33}}{2}\)
5. \(-\frac{5}{2} + \frac{\sqrt{33}}{2}\)
We check each one against our solutions:
- \(\frac{5}{2} + \frac{\sqrt{33}}{4}\) does not match either solution.
- \(\frac{5}{2} + \frac{\sqrt{33}}{2}\) does not match either solution.
- \(\frac{5}{2} - \frac{\sqrt{33}}{2}\) does not match either solution.
- \(-\frac{5}{2} - \frac{\sqrt{33}}{2}\) matches \(-\frac{5}{2} - \frac{\sqrt{33}}{2}\).
- \(-\frac{5}{2} + \frac{\sqrt{33}}{2}\) matches \(-\frac{5}{2} + \frac{\sqrt{33}}{2}\).
Therefore, the solutions to the equation \(x^2 + 5x = 2\) are:
[tex]\[
-\frac{5}{2} - \frac{\sqrt{33}}{2} \quad \text{and} \quad -\frac{5}{2} + \frac{\sqrt{33}}{2}
\][/tex]
These match the options:
- \(-\frac{5}{2} - \frac{\sqrt{33}}{2}\)
- [tex]\(-\frac{5}{2} + \frac{\sqrt{33}}{2}\)[/tex]
