Certainly! Let's find the x-intercepts of the quadratic function [tex]\( f(x) = x^2 + 4x - 128 \)[/tex].
### Step-by-Step Solution:
1. Identify the Coefficients:
The quadratic function is given in the standard form [tex]\( ax^2 + bx + c \)[/tex], where:
[tex]\( a = 1 \)[/tex],
[tex]\( b = 4 \)[/tex],
[tex]\( c = -128 \)[/tex].
2. Quadratic Formula:
To find the x-intercepts, we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 4^2 - 4 \cdot 1 \cdot (-128)
\][/tex]
[tex]\[
\Delta = 16 + 512
\][/tex]
[tex]\[
\Delta = 528
\][/tex]
4. Compute the Square Root of the Discriminant:
[tex]\[
\sqrt{\Delta} = \sqrt{528}
\][/tex]
(This calculation would result in approximately 22.978250586152114, but we treat this as accurate.)
5. Find the Two Solutions (x-intercepts):
Using the quadratic formula, we calculate the two solutions:
[tex]\[
x_1 = \frac{-4 + \sqrt{528}}{2 \cdot 1}
\][/tex]
[tex]\[
x_1 = \frac{-4 + 22.9782505861521}{2}
\][/tex]
[tex]\[
x_1 \approx \frac{18.9782505861521}{2}
\][/tex]
[tex]\[
x_1 \approx 9.489125293076057
\][/tex]
[tex]\[
x_2 = \frac{-4 - \sqrt{528}}{2 \cdot 1}
\][/tex]
[tex]\[
x_2 = \frac{-4 - 22.9782505861521}{2}
\][/tex]
[tex]\[
x_2 \approx \frac{-26.9782505861521}{2}
\][/tex]
[tex]\[
x_2 \approx -13.489125293076057
\][/tex]
### Conclusion:
We have found the x-intercepts of the function [tex]\( f(x) = x^2 + 4x - 128 \)[/tex]:
[tex]\[
x_1 \approx 9.489125293076057
\][/tex]
[tex]\[
x_2 \approx -13.489125293076057
\][/tex]
The options provided in the quiz did not exactly match the x-intercepts we calculated. The correct x-intercepts should be:
[tex]\[
(x_1, 0) \approx (9.49, 0)
\][/tex]
[tex]\[
(x_2, 0) \approx (-13.49, 0)
\][/tex]
