Solve the Quadratic Equation by Factorization: x − 1/x = 3

Question:

$$ x-\frac{1}{x}=3, \quad x\ne0 $$

Solution

$$ x^2-1=3x $$

$$ x^2-3x-1=0 $$

$$ x^2-\frac{3+\sqrt{13}}{2}x+\frac{3+\sqrt{13}}{2}x-1=0 $$

$$ \left(x-\frac{3+\sqrt{13}}{2}\right)\left(x-\frac{3-\sqrt{13}}{2}\right)=0 $$

Therefore,

$$ x=\frac{3+\sqrt{13}}{2} \quad \text{or} \quad x=\frac{3-\sqrt{13}}{2} $$

Hence,

$$ \boxed{x=\frac{3+\sqrt{13}}{2},\ \frac{3-\sqrt{13}}{2}} $$