What is an x-intercept of the quadratic function [tex]f(x)=(x-4)(x+2)[/tex]?
A. (-4,0)
B. (-2,0)
C. (0,2)
D. (4,-2)

To find the x-intercepts, set f ( x ) = 0 .
Solve the equation ( x − 4 ) ( x + 2 ) = 0 .
Find the solutions x = 4 and x = − 2 .
The x-intercepts are ( − 2 , 0 ) and ( 4 , 0 ) , so the answer is ( − 2 , 0 ) ​ .

Explanation

Finding x-intercepts To find the x -intercepts of the quadratic function f ( x ) = ( x − 4 ) ( x + 2 ) , we need to find the values of x for which f ( x ) = 0 . This occurs when ( x − 4 ) ( x + 2 ) = 0 .

Solving for x This equation is satisfied if either x − 4 = 0 or x + 2 = 0 . Let's solve each of these equations separately.

Solving x-4 = 0 If x − 4 = 0 , then x = 4 . So, one x -intercept is at x = 4 , which corresponds to the point ( 4 , 0 ) .

Solving x+2 = 0 If x + 2 = 0 , then x = − 2 . So, another x -intercept is at x = − 2 , which corresponds to the point ( − 2 , 0 ) .

Identifying the x-intercepts Therefore, the x -intercepts of the quadratic function f ( x ) = ( x − 4 ) ( x + 2 ) are ( 4 , 0 ) and ( − 2 , 0 ) . From the given options, ( − 2 , 0 ) and ( 4 , 0 ) are the x -intercepts.


Examples
Understanding x-intercepts is crucial in various real-world applications. For instance, in physics, if you model the height of a projectile as a function of time with a quadratic equation, the x-intercepts represent the times when the projectile hits the ground. Similarly, in business, if you model profit as a function of the number of items sold, the x-intercepts represent the break-even points where the profit is zero. Knowing how to find x-intercepts helps in analyzing these scenarios and making informed decisions.

Answered by GinnyAnswer | 2025-07-04

To find the x-intercepts of f ( x ) = ( x − 4 ) ( x + 2 ) , we set the function to zero and solve for x. This yields the x-intercepts at points ( 4 , 0 ) and ( − 2 , 0 ) . The correct answer is ( − 2 , 0 ) .
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Answered by Anonymous | 2025-07-14