Find the value of $x+y$
where $x^3+y^3=9xy-27.$
I tried using $a^3+b^3=(a+b)(a^2+b^2−ab)$ and also $a^3+b^3=(a+b)^3−3ab(a+b)$
but couldn't find the answer. Please explain how do I solve these types of questions.
Answer given = 6
[Math] Find the value of $x+y$ knowing that $x^3+y^3=9xy-27$
algebra-precalculusfactoringpolynomialssubstitution
Best Answer
Let's just consider real situation. Let $u=x+y$, $v=xy$, then $u^3-3uv=9v-27$, i.e., $(u+3)(u^2-3u+9-3v)=0$. So one case is $u=-3$, another is $u\neq -3$ and $v=\frac{1}{3}(u^2-3u+9)$. Since $x$, $y$ are both real, $u^2\ge 4v$, i.e., $(u-6)^2\le 0$, then $u=6$.