Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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]
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]
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
