Proving that for every real $x$ there exists $y$ with $x+y^2\in\mathbb{Q}$ I'm having trouble with proving this question on my assignment:

For all real numbers $x$, there exists a real number $y$ so that $x + y^2$ is rational.

I'm not sure exactly how to prove or disprove this. I proved earlier that for all real numbers $x$, there exists some $y$ such that $x+y$ is rational by cases and i'm assuming this is similar but I'm having trouble starting.

Hint $\rm\ x\! +\! y^2 = q\in\mathbb Q\iff y^2 = q\!-\!x\:$ so choose a rational $\rm\:q\ge x\:$ then let $\rm\:y = \sqrt{q-x}.$
Remark $\ $ Note how writing it out equationally makes it clearer what we need to do to solve it. Just as for "word problems", the key is learning how to translate the problem into the correct algebraic (equational) representation - the rest is easy.
