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Quadratic Functions: Standard Form

What are the x-intercepts of the graphed function [tex]f(x) = x^2 + 4x - 128[/tex]?

A. [tex]\((0, 0), (2, 0)\)[/tex]
B. [tex]\((2, 16), (0, 12)\)[/tex]
C. [tex]\((6, 0), (2, 16), (2, 0)\)[/tex]
D. [tex]\((0, 12), (0, 0), (2, 0)\)[/tex]



Answer :

GinnyAnswer
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]

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