To find the zeros of the quadratic function [tex]\( f(x) = 8x^2 + 2x - 21 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
### Step-by-Step Solution:
1. Write the quadratic equation:
[tex]\[
8x^2 + 2x - 21 = 0
\][/tex]
2. Identify the coefficients:
[tex]\[
a = 8, \quad b = 2, \quad c = -21
\][/tex]
3. Calculate the discriminant:
The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the coefficients, we get:
[tex]\[
\Delta = 2^2 - 4(8)(-21) = 4 + 672 = 676
\][/tex]
4. Determine the possible solutions using the quadratic formula:
The quadratic formula for finding the roots of [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[
x = \frac{{-b \pm \sqrt{\Delta}}}{2a}
\][/tex]
Substituting [tex]\( a = 8 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( \Delta = 676 \)[/tex]:
[tex]\[
x = \frac{{-2 \pm \sqrt{676}}}{2 \times 8}
\][/tex]
[tex]\[
x = \frac{{-2 \pm 26}}{16}
\][/tex]
5. Calculate the roots:
- For the positive root:
[tex]\[
x_1 = \frac{{-2 + 26}}{16} = \frac{24}{16} = 1.5
\][/tex]
- For the negative root:
[tex]\[
x_2 = \frac{{-2 - 26}}{16} = \frac{-28}{16} = -1.75
\][/tex]
6. Convert the decimal roots to mixed numbers:
- [tex]\( x_1 = 1.5 \)[/tex] is equivalent to [tex]\( 1 \frac{1}{2} \)[/tex]
- [tex]\( x_2 = -1.75 \)[/tex] is equivalent to [tex]\( -1 \frac{3}{4} \)[/tex]
### Conclusion:
The zeros of the function [tex]\( f(x) = 8x^2 + 2x - 21 \)[/tex] are:
[tex]\[
-1 \frac{3}{4} \quad \text{and} \quad 1 \frac{1}{2}
\][/tex]
So, the correct answer is:
[tex]\[
-1 \frac{3}{4} \quad \text{and} \quad 1 \frac{1}{2}
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{-1 \frac{3}{4} \text{ and } 1 \frac{1}{2}}
\][/tex]
