Let's carefully work through the problem:
Given the relationships [tex]\( \gamma^2 + \beta^2 = 15 \)[/tex] and [tex]\( \gamma \beta = 3 \)[/tex], we want to form a quadratic equation whose roots are [tex]\( \beta - 2 \)[/tex] and [tex]\( \beta - 2 \)[/tex].
1. Understand the System of Equations:
- We have two key equations:
[tex]\[
\gamma^2 + \beta^2 = 15
\][/tex]
[tex]\[
\gamma \beta = 3
\][/tex]
2. Solve for [tex]\( \gamma \)[/tex] and [tex]\( \beta \)[/tex]:
- First, solve the system of equations to find the possible values for [tex]\( \gamma \)[/tex] and [tex]\( \beta \)[/tex].
3. Calculate the Roots of the New Quadratic Equation:
- Let the roots be [tex]\( \beta - 2 \)[/tex] and [tex]\( \beta - 2 \)[/tex].
4. Formation of the Equation:
- Use the sum and product of these roots to form the quadratic equation.
Let's list out the solutions for [tex]\( \gamma \)[/tex] and [tex]\( \beta \)[/tex]:
The solutions for [tex]\( (\gamma, \beta) \)[/tex] are:
[tex]\[
(\gamma, \beta) = \left( -\frac{\sqrt{21}}{2} - \frac{3}{2}, \frac{3}{2} + \frac{\sqrt{21}}{2} \right), \left( \frac{3}{2} - \frac{\sqrt{21}}{2}, \frac{3}{2} + \frac{\sqrt{21}}{2} \right), \left( -\frac{\sqrt{21}}{2} - \frac{3}{2}, \frac{3}{2} - \frac{\sqrt{21}}{2} \right), \left( -\frac{\sqrt{21}}{2} + \frac{3}{2}, -\frac{\sqrt{21}}{2} + \frac{3}{2} \right)
\][/tex]
To simplify, let's consider any one pair of solutions, say [tex]\( \beta = \frac{3}{2} + \frac{\sqrt{21}}{2} \)[/tex].
The roots of the quadratic equation are [tex]\( \beta - 2 \)[/tex], which in this case would be:
[tex]\[
\beta - 2 = \frac{3}{2} + \frac{\sqrt{21}}{2} - 2 = -\frac{1}{2} + \frac{\sqrt{21}}{2}
\][/tex]
Therefore, if we consider all simplified valid roots, we have:
[tex]\[
-\frac{7}{2} + \frac{\sqrt{21}}{2}, -\frac{1}{2} - \frac{\sqrt{21}}{2}, -\frac{1}{2} + \frac{\sqrt{21}}{2}
\][/tex]
5. Sum and Product of the Roots:
- The sum of the roots is:
[tex]\[
-\left(\frac{7}{2} - \frac{\sqrt{21}}{2} \right)+ -\frac{\sqrt{21}}{2} + 1/2 - 1/2 = -8
\][/tex]
- The product of the roots is:
[tex]\[
\left(-\frac{7}{2} + \frac{\sqrt{21}}{2}\right) \times \left(\frac{-\sqrt{21}}{2} - 1/2\right)
= \left(-\frac{1- 1/2}+ \frac{\sqrt{21}}{2}\right)
\][/tex]
6. Forming the Final Quadratic Equation:
- Using the sum and product of the roots, the quadratic equation:
[tex]\[
x^2 - (sum \, of \, roots)x + (product \, of \, roots) = 0
\][/tex]
- Plug in the computed values:
[tex]\[
x^2 + 8x + \left(-\frac{7}{2} + \frac{\sqrt{21}}{2}\right) = 0
\][/tex]
So, putting everything together, the quadratic equation with the given conditions and the roots [tex]\( \beta - 2 \)[/tex] is:
[tex]\[
\boxed{x^2 + 8x + \left(\left(-\frac{7}{2} - \frac{\sqrt{21}}{2}\right)\right) = 0}
\][/tex]
