What are the x-intercepts of the parabolic function `f (x) = x^(2) -7x +81/4` ?

Options

A. `1 and 8 (1)/(4)`
B. `1 (1)/(4) and 5 (3)/(4)`
C. `1 (1)/(4) and 5 (3)/(4)`
D. `1 (1)/(2) and 5 (1)/(2)`

Answer

D

Explanation

The x-intercepts of a function are the values of x that make the function equal to 0. Factoring this function would be combersome because you would need to determine two numbers whose product is `8 (1)/(4)` and whose sum is -7. This is good candidate for completing the square.
Start with `x ^(2) - 7x + 8 (1)/(4) =0,` then subtract `8 (1)/(4)` from each side to get `x ^(2) -7x =-8 (1)/(4).` Half of the b coefficient, `-7,` is `-3 (1)/(2),` which, when squared, is `12 (1)/(4).` Add that to both sides to get `x ^(2) -7x + 12 (1)/(4)=4.` So `(x -3 (1)/(2))^(2)=4.` Take the square root of both sides to get` x -3 (1)/(2) = pm2.` Therefore, the x-intercepts are `3 (1)/(2) pm 2, or 1 (1)/(2) and 5 (1)/(2).` (D) is correct.