Answer :
Let's begin by matching each quadratic equation with its solution set.
We use the provided solutions:
1. [tex]\(2x^2 - 8x + 5 = 0\)[/tex]
- Solution: [tex]\[ [2 - \frac{\sqrt{6}}{2}, 2 + \frac{\sqrt{6}}{2}] \][/tex]
2. [tex]\(2x^2 - 10x - 3 = 0\)[/tex]
- Solution: [tex]\[ [\frac{5}{2} - \frac{\sqrt{31}}{2}, \frac{5}{2} + \frac{\sqrt{31}}{2}] \][/tex]
3. [tex]\(2x^2 - 8x - 3 = 0\)[/tex]
- Solution: [tex]\[ [2 - \frac{\sqrt{22}}{2}, 2 + \frac{\sqrt{22}}{2}] \][/tex]
4. [tex]\(2x^2 - 9x - 1 = 0\)[/tex]
- Solution: [tex]\[ [\frac{9}{4} - \frac{\sqrt{89}}{4}, \frac{9}{4} + \frac{\sqrt{89}}{4}] \][/tex]
5. [tex]\(2x^2 - 9x + 6 = 0\)[/tex]
- Solution: [tex]\[ [\frac{9}{4} - \frac{\sqrt{33}}{4}, \frac{9}{4} + \frac{\sqrt{33}}{4}] \][/tex]
Now let's identify the expressions provided:
- [tex]\(\frac{2 \pm \sqrt{80}}{4}\)[/tex]
- Simplification: [tex]\(\frac{2 \pm \sqrt{80}}{4} = \frac{2 \pm \sqrt{4 \cdot 20}}{4} = \frac{2 \pm 2\sqrt{20}}{4} = \frac{2 \pm 2\sqrt{2 \cdot 10}}{4} = \frac{2 \pm 2\sqrt{10}}{4} = \frac{1 \pm \sqrt{10}}{2}\)[/tex]
- None of the equations match with this solution set.
- [tex]\(\frac{4 \pm \sqrt{22}}{2}\)[/tex]
- Simplification: [tex]\(\frac{4 \pm \sqrt{22}}{2} = 2 \pm \frac{\sqrt{22}}{2}\)[/tex]
- This matches with the equation [tex]\(2x^2 - 8x - 3 = 0\)[/tex].
Hence, the correct matches are:
[tex]\[ \begin{array}{l} \frac{2 \pm \sqrt{80}}{4} \longrightarrow \text{None of the provided equations} \\ \frac{4 \pm \sqrt{22}}{2} \longrightarrow 2x^2 - 8x - 3 = 0 \\ \end{array} \][/tex]
We use the provided solutions:
1. [tex]\(2x^2 - 8x + 5 = 0\)[/tex]
- Solution: [tex]\[ [2 - \frac{\sqrt{6}}{2}, 2 + \frac{\sqrt{6}}{2}] \][/tex]
2. [tex]\(2x^2 - 10x - 3 = 0\)[/tex]
- Solution: [tex]\[ [\frac{5}{2} - \frac{\sqrt{31}}{2}, \frac{5}{2} + \frac{\sqrt{31}}{2}] \][/tex]
3. [tex]\(2x^2 - 8x - 3 = 0\)[/tex]
- Solution: [tex]\[ [2 - \frac{\sqrt{22}}{2}, 2 + \frac{\sqrt{22}}{2}] \][/tex]
4. [tex]\(2x^2 - 9x - 1 = 0\)[/tex]
- Solution: [tex]\[ [\frac{9}{4} - \frac{\sqrt{89}}{4}, \frac{9}{4} + \frac{\sqrt{89}}{4}] \][/tex]
5. [tex]\(2x^2 - 9x + 6 = 0\)[/tex]
- Solution: [tex]\[ [\frac{9}{4} - \frac{\sqrt{33}}{4}, \frac{9}{4} + \frac{\sqrt{33}}{4}] \][/tex]
Now let's identify the expressions provided:
- [tex]\(\frac{2 \pm \sqrt{80}}{4}\)[/tex]
- Simplification: [tex]\(\frac{2 \pm \sqrt{80}}{4} = \frac{2 \pm \sqrt{4 \cdot 20}}{4} = \frac{2 \pm 2\sqrt{20}}{4} = \frac{2 \pm 2\sqrt{2 \cdot 10}}{4} = \frac{2 \pm 2\sqrt{10}}{4} = \frac{1 \pm \sqrt{10}}{2}\)[/tex]
- None of the equations match with this solution set.
- [tex]\(\frac{4 \pm \sqrt{22}}{2}\)[/tex]
- Simplification: [tex]\(\frac{4 \pm \sqrt{22}}{2} = 2 \pm \frac{\sqrt{22}}{2}\)[/tex]
- This matches with the equation [tex]\(2x^2 - 8x - 3 = 0\)[/tex].
Hence, the correct matches are:
[tex]\[ \begin{array}{l} \frac{2 \pm \sqrt{80}}{4} \longrightarrow \text{None of the provided equations} \\ \frac{4 \pm \sqrt{22}}{2} \longrightarrow 2x^2 - 8x - 3 = 0 \\ \end{array} \][/tex]
