To solve the equation [tex]\(x^2 - 2z^2 + 5x - 30 = 0\)[/tex] for [tex]\(x\)[/tex], we need to follow a systematic approach. Here’s a detailed, step-by-step process to find the solutions for [tex]\(x\)[/tex]:
1. Rewrite the equation:
[tex]\[
x^2 - 2z^2 + 5x - 30 = 0
\][/tex]
2. Recall the quadratic formula:
The quadratic formula for solving an equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation [tex]\(x^2 - 2z^2 + 5x - 30 = 0\)[/tex]:
- [tex]\(a = 1\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(b = 5\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(c = -2z^2 - 30\)[/tex] (constant term)
3. Identify the coefficients and apply the formula:
Substitute [tex]\(a = 1\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(c = -2z^2 - 30\)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-2z^2 - 30)}}{2 \cdot 1}
\][/tex]
4. Simplify inside the square root:
Simplify the discriminant part of the formula:
[tex]\[
5^2 - 4 \cdot 1 \cdot (-2z^2 - 30) = 25 + 8z^2 + 120
\][/tex]
[tex]\[
25 + 8z^2 + 120 = 8z^2 + 145
\][/tex]
5. Complete the quadratic formula:
Now substitute back into the quadratic formula:
[tex]\[
x = \frac{-5 \pm \sqrt{8z^2 + 145}}{2}
\][/tex]
6. Separate the solutions:
The solutions for [tex]\(x\)[/tex] are thus:
[tex]\[
x_1 = \frac{-5 + \sqrt{8z^2 + 145}}{2}
\][/tex]
and
[tex]\[
x_2 = \frac{-5 - \sqrt{8z^2 + 145}}{2}
\][/tex]
7. Present the final solutions:
Therefore, the solutions for [tex]\(x\)[/tex] are:
[tex]\[
x = \frac{-5 \pm \sqrt{8z^2 + 145}}{2}
\][/tex]
Putting it explicitly, the solutions are:
[tex]\[
x_1 = \frac{-5 + \sqrt{8z^2 + 145}}{2}
\][/tex]
[tex]\[
x_2 = \frac{-5 - \sqrt{8z^2 + 145}}{2}
\][/tex]
So, the final definitive answers are:
[tex]\[
x = \left[-\frac{5}{2} - \frac{\sqrt{8z^2 + 145}}{2}, -\frac{5}{2} + \frac{\sqrt{8z^2 + 145}}{2}\right]
\][/tex]
