What is the [tex][tex]$x$[/tex][/tex]-intercept of the graph of the function [tex][tex]$f(x)=x^2-16x+64$[/tex][/tex]?

A. [tex][tex]$(-8,0)$[/tex][/tex]
B. [tex][tex]$(0,8)$[/tex][/tex]
C. [tex][tex]$(8,0)$[/tex][/tex]
D. [tex][tex]$(0,-8)$[/tex][/tex]



Answer :

To find the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = x^2 - 16x + 64 \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( f(x) = 0 \)[/tex].

1. Start with the function:
[tex]\[ f(x) = x^2 - 16x + 64 \][/tex]

2. Set [tex]\( f(x) \)[/tex] equal to 0:
[tex]\[ x^2 - 16x + 64 = 0 \][/tex]

3. Solve the quadratic equation:
Notice that we can factor the quadratic equation [tex]\( x^2 - 16x + 64 \)[/tex]:
[tex]\[ x^2 - 16x + 64 = (x - 8)^2 = 0 \][/tex]

4. Set the factored form equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 8)^2 = 0 \][/tex]

5. Take the square root of both sides:
[tex]\[ x - 8 = 0 \][/tex]

6. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 8 \][/tex]

Therefore, the [tex]\( x \)[/tex]-intercept is at [tex]\( x = 8 \)[/tex]. The coordinate of the [tex]\( x \)[/tex]-intercept is:
[tex]\[ (8, 0) \][/tex]

Thus, the correct answer is [tex]\( (8, 0) \)[/tex].