Certainly! Let's solve for [tex]$\alpha$[/tex] and [tex]$\beta$[/tex], the roots of the given quadratic equation [tex]\(2x^2 + 8x + 7 = 0\)[/tex].
### Finding the Roots
First, we apply the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the equation [tex]\(2x^2 + 8x + 7 = 0\)[/tex], we have:
[tex]\[ a = 2, \quad b = 8, \quad c = 7 \][/tex]
Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac = 8^2 - 4 \cdot 2 \cdot 7 = 64 - 56 = 8 \][/tex]
The roots are then:
[tex]\[ \alpha = \frac{-8 + \sqrt{8}}{2 \cdot 2} = \frac{-8 + 2\sqrt{2}}{4} = -2 + \frac{\sqrt{2}}{2} \][/tex]
[tex]\[ \beta = \frac{-8 - \sqrt{8}}{2 \cdot 2} = \frac{-8 - 2\sqrt{2}}{4} = -2 - \frac{\sqrt{2}}{2} \][/tex]
Numerically, these roots are approximately:
[tex]\[ \alpha \approx -1.2928932188134525 \][/tex]
[tex]\[ \beta \approx -2.7071067811865475 \][/tex]
### (i) Find [tex]\(\alpha^2 + \beta^2\)[/tex]
First, we compute [tex]\(\alpha^2\)[/tex] and [tex]\(\beta^2\)[/tex]:
[tex]\[ \alpha^2 \approx (-1.2928932188134525)^2 \approx 1.6715728752538102 \][/tex]
[tex]\[ \beta^2 \approx (-2.7071067811865475)^2 \approx 7.32842712474619 \][/tex]
Summing these:
[tex]\[ \alpha^2 + \beta^2 \approx 1.6715728752538102 + 7.32842712474619 = 9.0 \][/tex]
### (ii) Find [tex]\(\alpha^2 - \beta^2\)[/tex]
Subtracting:
[tex]\[ \alpha^2 - \beta^2 \approx 1.6715728752538102 - 7.32842712474619 \approx -5.65685424949238 \][/tex]
### Summary
So, the values are:
- [tex]\(\alpha^2 + \beta^2 \approx 9.0\)[/tex]
- [tex]\(\alpha^2 - \beta^2 \approx -5.65685424949238\)[/tex]
These are the detailed steps and calculations for finding the given values. The roots and results provided are accurate based on these operations.
