How to isolate $y$ when in form $y^2 + xy$

Question

I understand how to usually isolate it but the $x$ variable is leaving me confused. Like how would you isolate $y$, when it is $3x^3 + xy - x - 2y^2 = 2$. I can't seem to find any info online.

Answer 1

Bad news first: this is not a function in either $y$ or $x$.

What you can do is rearrange it in the form $a(x)y^2+b(x)y+c(x)=0$ and then use the quadratic formula; this gives two individual functions, one for the "positive" branch and one for the "negative" branch. Taking your example:

$$\begin{align} 3x^3+xy-x-2y^2&=2\\ -2y^2+xy+3x^3-x-2&=0 \end{align}$$ Extract the "coefficients": $$\begin{align} a(x)&=-2\\ b(x)&=x\\ c(x)&=3x^3-x-2 \end{align}$$ And plug into the quadratic formula. $$\begin{align} y(x)&=\frac{-b(x)\pm\sqrt{b(x)^2-4a(x)c(x)}}{2a(x)}\\ &=\frac{-(x)\pm\sqrt{(x)^2-4(-2)(3x^3-x-2)}}{2(-2)}\\ &=\frac{-x\pm\sqrt{x^2+24x^3-8x-8}}{-4}\\ &=\frac{x\pm\sqrt{24x^3+x^2-8x-8}}{4}\\ \end{align}$$ We can then split this up based on the sign we choose for the $\pm$ before the square root.