What is the midpoint of the $x$-intercepts of $f(x)=(x-2)(x-4)$?
Answer (1)
Find the x-intercepts by setting f ( x ) = ( x − 2 ) ( x − 4 ) = 0 , which gives x = 2 and x = 4 .
Identify the x-intercepts as points ( 2 , 0 ) and ( 4 , 0 ) .
Apply the midpoint formula: M i d p o in t = ( 2 2 + 4 , 2 0 + 0 ) .
Calculate the midpoint: M i d p o in t = ( 3 , 0 ) . The final answer is ( 3 , 0 ) .
Explanation
Find the x-intercepts The problem asks us to find the midpoint of the x-intercepts of the function f ( x ) = ( x − 2 ) ( x − 4 ) . First, we need to find the x-intercepts, which are the points where the function equals zero.
Solve for x To find the x-intercepts, we set f ( x ) = 0 and solve for x :
( x − 2 ) ( x − 4 ) = 0 This equation is satisfied when x − 2 = 0 or x − 4 = 0 . Solving these equations gives us x = 2 and x = 4 . So, the x-intercepts are the points ( 2 , 0 ) and ( 4 , 0 ) .
Apply the midpoint formula Now, we need to find the midpoint of the x-intercepts ( 2 , 0 ) and ( 4 , 0 ) . The midpoint formula is given by: M i d p o in t = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) Plugging in the coordinates of the x-intercepts, we get: M i d p o in t = ( 2 2 + 4 , 2 0 + 0 ) = ( 2 6 , 2 0 ) = ( 3 , 0 ) So, the midpoint of the x-intercepts is ( 3 , 0 ) .
State the final answer Therefore, the midpoint of the x-intercepts of f ( x ) = ( x − 2 ) ( x − 4 ) is ( 3 , 0 ) .
Examples
Understanding midpoints is crucial in various real-world scenarios. For instance, when designing a bridge, engineers need to find the midpoint to ensure balanced support. Similarly, in computer graphics, calculating midpoints helps in rendering shapes and creating smooth animations. This concept also applies in urban planning, where the midpoint between two locations can determine the optimal placement for a service or facility, ensuring equal accessibility for residents.
