Question

This following equation defines y as a function of x: X^3-27=8. What are the steps to solving this problem? How do I show my work? I was stuck after I got -2y=-x^3+8

Answer 1

okay, substract 8 from both sides

x^3-35=0

use the formula a^3-b^3=(a-b)(a^2+ab+b^2)

(x-(35)^(1/3))*(x^2+x*(35)^(1/3)+(35)^(2/3))=0

this leaves us with two options

(x-(35)^(1/3))=0, which yields x=(35)^(1/3)

or the other option (x^2+x*(35)^(1/3)+(35)^(2/3))=0

the function (x^2+x*(35)^(1/3)+(35)^(2/3)) has no real roots, since the discriminant is positive;

this leaves only the first option as correct
x=(35)^(1/3), where (35)^(1/3) is read as the third root of 35, or root 3 of 35
it is also written