Find the area of the space enclosed by points $(x, y)$ such that fullfill the following condition: $9x^3+9x^2y-45x^2=4y^3+4xy^2-20y^2$

Question

Find the area of the space enclosed by points $(x, y)$ such that fullfill the following condition: $9x^3+9x^2y-45x^2=4y^3+4xy^2-20y^2$

My solution goes as follows:

Factorizing we have:

$(9x^2-4y^2)(x+y-5)=0$

$(3x+2y)(3x-2y)(x+y-5)=0$

Hence we get that the vertices of the triangle are points $(-10, 15), (0, 0)$ and $(2, 3)$ and from this we conclude that the area is $30$.

My question is how was I supposed to think of the particular factorization? Can you propose an intuitive method?

Answer 1

Moving everything to one side, $9x^3+9x^2y-45x^2-4xy^2+20y^2-4y^3=0$. Thinking about factorizations, we see the appearance of multiple $9x$'s and $4y$'s. Take out the $9x^2$ and $4y^2$ first to get $9x^2(x+y-5)+4y^2(-x-y+5)$. This is the woah moment, and the rest follows easily.

Alternatively, you can do something similar to Simon's Favorite Factoring Trick (SFFT), which you can learn more about here: https://artofproblemsolving.com/wiki/index.php/Simon%27s_Favorite_Factoring_Trick.

Answer 2

Step by step, we have:

$$9(x^3+x^2y-5x^2)=4(y^3+xy^2-5y^2)$$ $$\Rightarrow 9x^2(x + y - 5) = 4y^2 (y + x - 5)$$

and moving everything to one side gives $(9x^2-4y^2)(x + y - 5) = 0$.