Quadratic Equation ( x and y ) formula

Question

There's a formula for solving a Quadratic Equation that has $x$ as a variable; e.g:

$$2x^2 + 5x - 8x = 0$$

Step 1 (is to find delta/discriminant):

$\Delta = b^2 - 4ac$

Step 2 (is to find $x_1$ and $x_2$):

$x$ = $\dfrac{-(b) \pm \sqrt\Delta}{2a}$

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My question is... Is it possible to solve an equation which has $x$ & $y$ variables, with the formula above? Or is there another formula for equations with $x$ & $y$ variables?

Example:

$$2x^2 + 5xy - 8y^2 = 0$$

Answer 1

HINT: if $$y=0$$ and $$x=0$$ are not a solutions of your equation you can write $$2\frac{x}{y}+5-8\frac{y}{x}=0$$ Setting $t=x/y$ you will have $$2t+5-\frac{8}{t}=0$$ multiplying by $t$: $$2t^2+5t-8=0$$

Answer 2

Going off of @lioness99a's comment, for your example you can set $a=2$, $b=5y$, and $c=-8y^2$ to get

\begin{align*} x &= \frac{-5y \pm \sqrt{(5y)^2 - 4\cdot 2 \cdot (-8y^2)}}{2\cdot 2} \\ &= \frac{-5y \pm \sqrt{89}|y|}{4} \\ &= \frac{1}{4} (-5 \pm \sqrt{89})y \end{align*}

or fix $x$ and treat $y$ as the variable.