To find the zeros of the quadratic function [tex]\( f(x) = 8x^2 - 16x - 15 \)[/tex], we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 8 \)[/tex], [tex]\( b = -16 \)[/tex], and [tex]\( c = -15 \)[/tex].
1. Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute [tex]\( a = 8 \)[/tex], [tex]\( b = -16 \)[/tex], and [tex]\( c = -15 \)[/tex]:
[tex]\[
\Delta = (-16)^2 - 4 \cdot 8 \cdot (-15)
\][/tex]
[tex]\[
\Delta = 256 + 480
\][/tex]
[tex]\[
\Delta = 736
\][/tex]
2. Find the square root of the discriminant:
[tex]\[
\sqrt{736}
\][/tex]
This is the value we need to evaluate the quadratic formula.
3. Apply the quadratic formula:
[tex]\[
x = \frac{-(-16) \pm \sqrt{736}}{2 \cdot 8}
\][/tex]
Simplify:
[tex]\[
x = \frac{16 \pm \sqrt{736}}{16}
\][/tex]
4. Separate into two solutions:
[tex]\[
x_1 = \frac{16 - \sqrt{736}}{16}
\][/tex]
[tex]\[
x_2 = \frac{16 + \sqrt{736}}{16}
\][/tex]
5. Simplify each solution:
[tex]\[
x_1 = 1 - \frac{\sqrt{736}}{16}
\][/tex]
[tex]\[
x_2 = 1 + \frac{\sqrt{736}}{16}
\][/tex]
Given the result:
- The discriminant [tex]\( \Delta = 736 \)[/tex]
- The approximate solutions are:
[tex]\[
x_1 \approx -0.695582495781317
\][/tex]
[tex]\[
x_2 \approx 2.6955824957813173
\][/tex]
Comparing these to the provided options:
- [tex]\( x=-1-\sqrt{2} \)[/tex] and [tex]\( x=-1+\sqrt{2} \)[/tex]
- [tex]\( x=-1-\sqrt{\frac{15}{8}} \)[/tex] and [tex]\( x=-1+\sqrt{\frac{15}{8}} \)[/tex]
- [tex]\( x=1-\sqrt{\frac{23}{8}} \)[/tex] and [tex]\( x=1+\sqrt{\frac{23}{8}} \)[/tex]
- [tex]\( x=1-\sqrt{7} \)[/tex] and [tex]\( x=1+\sqrt{7} \)[/tex]
The correct solution matches with:
[tex]\[
x = 1 - \sqrt{7} \quad \text{and} \quad x = 1 + \sqrt{7}
\][/tex]
Hence, the zeros of the quadratic function [tex]\( f(x) = 8x^2 - 16x - 15 \)[/tex] are:
[tex]\[
x = 1 - \sqrt{7} \quad \text{and} \quad x = 1 + \sqrt{7}
\][/tex]
