Two quadratic polynomial fraction sums up to be 8
Find all real solutions for: $$\frac{3x}{2x^2-7x+7} + \frac{27x}{2x^2+6x+7} = 8$$
My work is down in the Answers section.
Solution 1:
$$\text{Let }A=2x^2+6x+7$$ $$\frac{3x}{A-13x} + \frac{27x}A = 8$$
$$3Ax + 27x(A-13x) = 8A(A-13x)$$ $$3Ax + 27Ax-351x = 8A^2-104Ax$$ $$8A^2-134Ax+351x = 0$$ $$(4A-13x)(2A-27x)=0$$
$$4A-13x=0$$ $$8x^2+24x+28-13x = 0$$ $$8x^2+11x+28=0$$ $$\Delta<0\implies\text{No real solutions}$$
$$2A-27x=0$$ $$4x^2+12x+14-27x=0$$ $$4x^2-15x+14=0\implies x=2, \frac74$$