Which point is an $x$-intercept of the quadratic function $f(x)=(x+6)(x-3)$?
A. $(0,-6)$
B. $(6,0)$
C. $(-6,0)$
D. $(0,6)$
Answer (1)
Set the quadratic function to zero: ( x + 6 ) ( x − 3 ) = 0 .
Solve for x by setting each factor to zero: x + 6 = 0 or x − 3 = 0 , which gives x = − 6 or x = 3 .
The x -intercepts are the points where y = 0 , so the x -intercepts are ( − 6 , 0 ) and ( 3 , 0 ) .
Identify the correct x -intercept from the given options: ( − 6 , 0 ) .
Explanation
Finding the x-intercept To find the x -intercept of the quadratic function f ( x ) = ( x + 6 ) ( x − 3 ) , we need to find the points where the function's value is zero. In other words, we need to solve the equation f ( x ) = 0 .
Setting the function to zero We set the function equal to zero: ( x + 6 ) ( x − 3 ) = 0
Solving for x This equation is already factored, which makes it easy to find the solutions. We have two factors: ( x + 6 ) and ( x − 3 ) . If either of these factors is equal to zero, then the entire product is equal to zero.
So, we set each factor equal to zero and solve for x :
x + 6 = 0 x = − 6
x − 3 = 0 x = 3
Identifying the x-intercepts The x -intercepts are the points where y = 0 . Therefore, the x -intercepts are ( − 6 , 0 ) and ( 3 , 0 ) .
Checking the options Now we check which of the given options is one of the x -intercepts we found. The options are:
( 0 , − 6 ) ( 6 , 0 ) ( − 6 , 0 ) ( 0 , 6 )
The point ( − 6 , 0 ) matches one of our x -intercepts.
Examples
Understanding x-intercepts is crucial in various real-world applications. For instance, if you're modeling the trajectory of a ball thrown in the air with a quadratic function, the x-intercepts would represent the points where the ball hits the ground. Similarly, in business, if you're modeling profit as a function of sales, the x-intercepts would represent the break-even points where the profit is zero. Knowing how to find x-intercepts helps in predicting outcomes and making informed decisions in these scenarios.
