Solve the equation, find the x

Question

I need to find the x in thid equation. How is it done?

a = x+1/x

I've tried turning it into x² = 1-x times a, but it's not a system.. so.. any ideas?

Answer 1

$$\begin{align}a=x+\frac 1x&\Rightarrow ax=x^2+1\\&\Rightarrow x^2-ax+1=0\\&\Rightarrow x=\frac{a\pm\sqrt{a^2-4}}{2}.\end{align}$$ Here, note that for $A\not=0$, $$Ax^2+Bx+C=0\Rightarrow x=\frac{-B\pm\sqrt{B^2-4AC}}{2A}.$$

Answer 2

Going to the Quadratic Formula is the natural move. Alternately, note that $$\left(x-\frac{1}{x}\right)^2=\left(x+\frac{1}{x}\right)^2-4=a^2-4.$$ It follows that $$x-\frac{1}{x}=\pm\sqrt{a^2-4}.$$ But we have $$x+\frac{1}{x}=a.$$ Add, divide by $2$.