Answer:
Step 1: Rearrange the Equation
First, let's multiply the entire equation by $-2$ to eliminate the fraction:
[tex]{-2(-8) + -2(5x) + -2\left(-\frac{1}{2}x^2\right) = 0}[/tex]
This simplifies to:
[tex]16 - 10x + x^2 = 0.[/tex]
Now, we can rewrite it in standard form:
[tex]x^2 - 10x + 16 = 0.[/tex]
Step 2: Identify Coefficients
Here, we have:
- a = 1
- b = -10
- c = 16
Step 3: Use the Quadratic Formula
The quadratic formula is given by:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.[/tex]
Step 4: Calculate the Discriminant
First, we calculate the discriminant
[tex]b^2 - 4ac[/tex]
[tex]{b^2 - 4ac = (-10)^2 - 4(1)(16) = 100 - 64 = 36.}[/tex]
Step 5: Substitute into the Quadratic Formula
Now, substitute a , b , and the discriminant into the quadratic formula:
[tex]x = \frac{-(-10) \pm \sqrt{36}}{2(1)} = \frac{10 \pm 6}{2}.[/tex]
Step 6: Calculate the Two Possible Solutions
Now we calculate the two possible values for x :
1.
[tex]x_1 = \frac{10 + 6}{2} = \frac{16}{2} = 8[/tex]
2.
[tex]x_2 = \frac{10 - 6}{2} = \frac{4}{2} = 2[/tex]
Conclusion
The solutions to the quadratic equation are
[tex] \text{ \boxed{x = 8} \quad \text{and} \quad \boxed{ x = 2}}[/tex]
