Final answer to the problem
$y=-\frac{2}{3}x$
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Step-by-step Solution
Specify the solving method
1
Find the break even points of the polynomial $\frac{4x^2+12yx+9y^2}{3y+2x}$ by putting it in the form of an equation and then set it equal to zero
$\frac{4x^2+12yx+9y^2}{3y+2x}=0$
2
The trinomial $4x^2+12yx+9y^2$ is a perfect square trinomial, because it's discriminant is equal to zero
$\Delta=b^2-4ac=12^2-4\left(4\right)\left(9\right) = 0$
3
Using the perfect square trinomial formula
$a^2+2ab+b^2=(a+b)^2,\:where\:a=\sqrt{4x^2}\:and\:b=\sqrt{9y^2}$
4
Factoring the perfect square trinomial
$\frac{\left(2x+3y\right)^{2}}{3y+2x}=0$
5
Simplify the fraction $\frac{\left(2x+3y\right)^{2}}{3y+2x}$ by $3y+2x$
$3y+2x=0$
6
We need to isolate the dependent variable , we can do that by simultaneously subtracting $2x$ from both sides of the equation
$3y=-2x$
7
Simplify, dividing both sides of the equality by
$y=-\frac{2}{3}x$
Final answer to the problem
$y=-\frac{2}{3}x$