To determine the rational zeros of the quadratic function [tex]\( f(x) = 5x^2 - 35x - 300 \)[/tex], follow these steps:
1. Set the function equal to zero:
[tex]\[
5x^2 - 35x - 300 = 0
\][/tex]
We're solving for [tex]\( x \)[/tex] such that the function equals zero.
2. Identify the quadratic formula:
The quadratic formula to solve [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the function [tex]\( 5x^2 - 35x - 300 \)[/tex], the coefficients are:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = -35 \)[/tex]
- [tex]\( c = -300 \)[/tex]
3. Compute the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is calculated as:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
[tex]\[
\Delta = (-35)^2 - 4(5)(-300)
\][/tex]
[tex]\[
\Delta = 1225 + 6000
\][/tex]
[tex]\[
\Delta = 7225
\][/tex]
4. Simplify the quadratic formula:
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\(\sqrt{\Delta}\)[/tex] back into the quadratic formula:
[tex]\[
x = \frac{-(-35) \pm \sqrt{7225}}{2 \cdot 5}
\][/tex]
[tex]\[
x = \frac{35 \pm 85}{10}
\][/tex]
5. Solve for the zeros:
- First zero:
[tex]\[
x = \frac{35 + 85}{10} = \frac{120}{10} = 12
\][/tex]
- Second zero:
[tex]\[
x = \frac{35 - 85}{10} = \frac{-50}{10} = -5
\][/tex]
Therefore, the rational zeros of the function [tex]\( f(x) = 5x^2 - 35x - 300 \)[/tex] are:
[tex]\[
\boxed{-5, 12}
\][/tex]
Given the options, the correct answer is:
[tex]\[
-5, 12
\][/tex]
