Solve the equation $\frac{x^2-10x+15}{x^2-6x+15}=\frac{4x}{x^2-12x+15}$ Solve the equation $$\dfrac{x^2-10x+15}{x^2-6x+15}=\dfrac{4x}{x^2-12x+15}.$$
First we have $$x^2-6x+15\ne0$$ which is true for every $x$ ($D_1=k^2-ac=9-15<0$) and $$x^2-12x+15\ne0\Rightarrow x\ne6\pm\sqrt{21}.$$ Now $$(x^2-10x+15)(x^2-12x+15)=4x(x^2-6x+15)\\x^4-12x^3+15x^2-10x^3+120x-150x+15x^2-180x+225=\\=4x^3-24x^2+60x$$ which is an equation I can't solve. I tried to simplify the LHS by $$\dfrac{x^2-10x+15}{x^2-6x+15}=\dfrac{(x^2-6x+15)-4x}{x^2-6x+15}=1-\dfrac{4x}{x^2-6x+15}$$ but this isn't helpful at all. Any help would be appreciated! :) Thank you in advance!

Componendo and dividendo yields:
$$\frac{x^2-6x+15}{x^2-10x+15} = \frac{x^2-12x+15}{4x}$$
$$\Rightarrow \frac{4x}{x^2 - 10x + 15} = \frac{x^2-16x+15}{4x} \tag{$\frac{a-b}{b} = \frac{c-d}{d}$}$$
$$\Rightarrow u=x^2-13x+15: \frac{4x}{u + 3x} = \frac{u - 3x}{4x}$$
$$\Rightarrow u^2 - 9x^2 = 16x^2$$
$$\Rightarrow (u - 5x)(u + 5x) = 0$$
$$\Rightarrow (x^2 - 18x + 15)(x^2 - 8x + 15) = 0$$
