Prove that there are no rational solutions to the equation x^3 + x^2 = 1
Let $\displaystyle x=\frac{p}{q}$ with $\displaystyle p\wedge q=1$; you get $\displaystyle p^3+qp^2=q^3$ hence (...) $\displaystyle q|p$ and thus $\displaystyle q=1$, which mean $\displaystyle x$ is an integer. I let you fill in the (quick) dots and handle it from there (beware, $\displaystyle x^3$ may be negative).