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What are the [tex]$x$[/tex]-intercepts of the graph of the function [tex]$f(x)=x^2+4x-12$[/tex]?

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



Answer :

GinnyAnswer
To find the [tex]\(x\)[/tex]-intercepts of the function [tex]\(f(x) = x^2 + 4x - 12\)[/tex], we need to determine the values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 0\)[/tex].

This involves solving the equation:
[tex]\[x^2 + 4x - 12 = 0\][/tex]

To solve this quadratic equation, we use the quadratic formula:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]

Here, [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -12\)[/tex].

First, we calculate the discriminant:
[tex]\[Discriminant = b^2 - 4ac\][/tex]
[tex]\[Discriminant = 4^2 - 4 \cdot 1 \cdot (-12)\][/tex]
[tex]\[Discriminant = 16 + 48\][/tex]
[tex]\[Discriminant = 64\][/tex]

Next, we use the quadratic formula to find the two solutions:
[tex]\[x_1 = \frac{-b + \sqrt{Discriminant}}{2a}\][/tex]
[tex]\[x_1 = \frac{-4 + \sqrt{64}}{2 \cdot 1}\][/tex]
[tex]\[x_1 = \frac{-4 + 8}{2}\][/tex]
[tex]\[x_1 = \frac{4}{2}\][/tex]
[tex]\[x_1 = 2\][/tex]

[tex]\[x_2 = \frac{-b - \sqrt{Discriminant}}{2a}\][/tex]
[tex]\[x_2 = \frac{-4 - \sqrt{64}}{2 \cdot 1}\][/tex]
[tex]\[x_2 = \frac{-4 - 8}{2}\][/tex]
[tex]\[x_2 = \frac{-12}{2}\][/tex]
[tex]\[x_2 = -6\][/tex]

Thus, the [tex]\(x\)[/tex]-intercepts of the graph are [tex]\((2, 0)\)[/tex] and [tex]\((-6, 0)\)[/tex].

From the provided choices, the correct answer is:
[tex]\[ (-6,0), (2,0) \][/tex]

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