To find the zeros of the quadratic function [tex]\( f(x) = 8x^2 - 16x - 15 \)[/tex], we will solve the equation [tex]\( 8x^2 - 16x - 15 = 0 \)[/tex] using the quadratic formula. The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
Given that:
[tex]\[
a = 8, \quad b = -16, \quad c = -15
\][/tex]
Step 1: Calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = (-16)^2 - 4 \cdot 8 \cdot (-15) = 256 + 480 = 736
\][/tex]
Step 2: Compute the two solutions for [tex]\( x \)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\( b \)[/tex], [tex]\( \Delta \)[/tex], and [tex]\( a \)[/tex]:
[tex]\[
x = \frac{-(-16) \pm \sqrt{736}}{2 \cdot 8} = \frac{16 \pm \sqrt{736}}{16}
\][/tex]
Step 3: Simplify the expression:
[tex]\[
x = \frac{16 \pm \sqrt{736}}{16}
\][/tex]
The two solutions (roots) are:
[tex]\[
x_1 = \frac{16 + \sqrt{736}}{16} \approx 2.6955824957813173
\][/tex]
[tex]\[
x_2 = \frac{16 - \sqrt{736}}{16} \approx -0.695582495781317
\][/tex]
Therefore, the zeros of the quadratic function [tex]\( f(x) = 8x^2 - 16x - 15 \)[/tex] are approximately:
[tex]\[
x \approx 2.6956 \quad \text{and} \quad x \approx -0.6956
\][/tex]
Among the provided options, none directly match the approximate numerical values of the zeros. The correct roots are:
[tex]\[
x \approx 2.6956 \quad \text{and} \quad x \approx -0.6956
\][/tex]
