Solve for the roots of the following equation.
x2/3 + x1/3 + 1 = 0
Is there some way to express this as a quadratic equation?
Create a new variable, u, and set it equal to x1/3.
u2 + u + 1 = 0
This new equation is quadratic.
The equation isn't factorable, so we use the quadratic formula to find the roots.
These are the roots for the new, u-based equation.
u = x1/3
We were right; this is a super nasty problem. Let's go ahead and check our results, though.
Luckily, most of the exponents cancel each other out.
And the other. That means our answers are and . It's really satisfying to get a long problem like this right after all this work. It gives us a warm, fuzzy feeling all over.