Quadratic Equation roots when there is a constant

Question

What are the roots of x2+x-2n where n is a constant? Can someone help me with a step by step approach?

Answer 1

$$\begin{align}0=x^2+x-2n&=x^2+x+\frac{1}{4}-\left(2n+\frac{1}{4}\right)\\&=\left(x+\frac{1}{2}\right)^2-\left(2n+\frac{1}{4}\right)\end{align}$$

Therefore, the equation is equivalent to $$\left(x+\frac{1}{2}\right)^2=2n+\frac{1}{4}$$

From where $$x=\pm\sqrt{2n+\frac{1}{4}}-\frac{1}{2}$$