To solve the quadratic equation
[tex]\[ -\frac{1}{2} x^2 + 5x = 8, \][/tex]
let's first rewrite it in standard form, which is
[tex]\[ ax^2 + bx + c = 0. \][/tex]
Subtract 8 from both sides of the equation:
[tex]\[ -\frac{1}{2} x^2 + 5x - 8 = 0. \][/tex]
Here, we can identify the coefficients:
[tex]\[ a = -\frac{1}{2}, \quad b = 5, \quad c = -8. \][/tex]
Next, we will use the quadratic formula to find the solutions for \( x \):
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \][/tex]
1. Calculate the discriminant (\( \Delta \)):
[tex]\[
\Delta = b^2 - 4ac.
\][/tex]
Plugging in the values, we get:
[tex]\[
\Delta = 5^2 - 4 \left(-\frac{1}{2}\right)(-8).
\][/tex]
Simplify the expression:
[tex]\[
\Delta = 25 - 4 \left(-\frac{1}{2}\right)(-8) = 25 - 16 = 9.
\][/tex]
2. Determine the square root of the discriminant:
[tex]\[
\sqrt{\Delta} = \sqrt{9} = 3.
\][/tex]
3. Apply the quadratic formula for the two possible values of \( x \):
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}.
\][/tex]
So we have:
[tex]\[
x = \frac{-5 \pm 3}{2 \left(-\frac{1}{2}\right)}.
\][/tex]
4. Calculate the two solutions separately:
[tex]\[
x_1 = \frac{-5 + 3}{-1} = \frac{-2}{-1} = 2,
\][/tex]
[tex]\[
x_2 = \frac{-5 - 3}{-1} = \frac{-8}{-1} = 8.
\][/tex]
Therefore, the values of \(x\) that satisfy the equation
[tex]\[ -\frac{1}{2} x^2 + 5x = 8 \][/tex]
are
[tex]\[ x = 2 \][/tex]
and
[tex]\[ x = 8. \][/tex]
So among the given options, the solutions are:
- \(2\)
- \(8\)
The other options (-8, -2, -1.5, 11.5) do not satisfy the equation.
