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

Set the function equal to zero: x 2 + 4 x − 12 = 0 .
Factor the quadratic equation: ( x + 6 ) ( x − 2 ) = 0 .
Solve for x by setting each factor to zero: x = − 6 or x = 2 .
The x -intercepts are ( − 6 , 0 ) , ( 2 , 0 ) ​ .

Explanation

Problem Analysis We are given the function f ( x ) = x 2 + 4 x − 12 and we want to find the x -intercepts of its graph. The x -intercepts are the points where the graph intersects the x -axis, which means f ( x ) = 0 . So, we need to solve the equation x 2 + 4 x − 12 = 0 for x .

Factoring the Quadratic To solve the quadratic equation x 2 + 4 x − 12 = 0 , we can try to factor it. We are looking for two numbers that multiply to -12 and add to 4. These numbers are 6 and -2, since 6 × ( − 2 ) = − 12 and 6 + ( − 2 ) = 4 . Therefore, we can factor the quadratic as ( x + 6 ) ( x − 2 ) = 0 .

Solving for x Now, we set each factor equal to zero and solve for x :


x + 6 = 0 or x − 2 = 0
Solving these equations gives us:
x = − 6 or x = 2

Finding the x-intercepts The x -intercepts are the points where f ( x ) = 0 , so the x -intercepts are ( − 6 , 0 ) and ( 2 , 0 ) .

Examples
Understanding x-intercepts is crucial in many real-world applications. For example, if you're modeling the trajectory of a ball thrown in the air with the function f ( x ) = x 2 + 4 x − 12 , the x-intercepts tell you where the ball lands (i.e., when the height f ( x ) is zero). Similarly, in business, if f ( x ) represents the profit of a company as a function of the number of units sold ( x ), the x-intercepts represent the break-even points where the company neither makes a profit nor incurs a loss. Finding these points helps in making informed decisions.

Answered by GinnyAnswer | 2025-07-05