How to find maximum and minimum value of a variable in 2 variable equation

Question

I have been given a equation $$4x^2 + 12xy + 10y^2 -4y +3= 0$$ How can I find maximum value of $y$ is this intermixed variable equation ? I have not been Introduced to multi variable calculus . can this be done with partial differentiation ?

Answer 1

$$4x^2 + 12xy + 10y^2 -4y +3 = (2x+3y)^2+(y-1)^2-1$$ then $$(2x+3y)^2+(y-1)^2=1$$ then $$-1\leq \mid y-1\mid\leq1$$ then $$y\leq 2$$ and $(x,y)=(-3,2)$ is an answer.

Answer 2

$$0= 4x^2+12xy+10y^2−4y+3 = (2x+3y)^2 + y^2 - 4y + 3 + 1 - 1=\\=(2x+3y)^2 + (y-2)^2 - 1 $$ So, you have $(2x+3y)^2 + (y-2)^2 - 1$. Then if you define two new independent variables , you can obtain :

$$z^2 + w^2 = 1$$

where $|w|<=1$, this means y belongs to [1;3].