To solve the equation
[tex]\[
\frac{x^2}{20 - 3x} = \frac{1}{2},
\][/tex]
we can proceed as follows:
1. Cross-multiply the given equation to eliminate the fraction:
[tex]\[
2 x^2 = 20 - 3x.
\][/tex]
2. Rearrange the equation to form a standard quadratic equation:
[tex]\[
2 x^2 + 3x - 20 = 0.
\][/tex]
3. This is a quadratic equation in the form [tex]\(ax^2 + bx + c = 0\)[/tex], with [tex]\(a = 2\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(c = -20\)[/tex].
4. To solve the quadratic equation, we can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
\][/tex]
5. Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-20)}}{2 \cdot 2}.
\][/tex]
6. Simplify under the square root:
[tex]\[
x = \frac{-3 \pm \sqrt{9 + 160}}{4}.
\][/tex]
[tex]\[
x = \frac{-3 \pm \sqrt{169}}{4}.
\][/tex]
[tex]\[
x = \frac{-3 \pm 13}{4}.
\][/tex]
7. This gives us two solutions when performing the calculation with each sign:
- For the positive square root:
[tex]\[
x = \frac{-3 + 13}{4} = \frac{10}{4} = 2.5.
\][/tex]
- For the negative square root:
[tex]\[
x = \frac{-3 - 13}{4} = \frac{-16}{4} = -4.
\][/tex]
8. Therefore, the possible values of [tex]\(x\)[/tex] are:
[tex]\[
x = 2.5 \quad \text{and} \quad x = -4.
\][/tex]
Thus, the solutions to the equation [tex]\(\frac{x^2}{20 - 3x} = \frac{1}{2}\)[/tex] are [tex]\(x = 2.5\)[/tex] and [tex]\(x = -4\)[/tex].
