Question

Solve for d. (d-e)/(3d +e)=d

Answer 1

$\dfrac{d-e}{3d+e}=d\ \ \ \ |\text{cross multiply}\\\\d(3d+e)=d-e\\\\3d^2+de=d-e\ \ \ |-d+e\\\\3d^2+de-d+e=0\\\\3d^2+(e-1)d+e=0$

Use the quadratic formula:

$a=3,\ b=e-1,\ c=e\\\\b^2=(e-1)^2-4(3)(e)=(e-1)^2-12e\\\\d=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\d=\dfrac{-(e-1)\pm\sqrt{(e-1)^2-12e}}{2\cdot3}\\\\\boxed{d=\dfrac{1-e\pm\sqrt{(e-1)^2-12e}}{6}}$