Question

Solve the equation: $x + \frac{1}{x} = 3$.

Answer 1

$\displaystyle x = \frac{3 \pm \sqrt{5}}{2}$

Answer 2

Step 1: Multiply both sides by $x$ (noting $x\neq0$):

$$x^2 + 1 = 3x.$$

Step 2: Rearrange into a standard quadratic:

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

Step 3: Apply the quadratic formula $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$:

$$x = \frac{3\pm\sqrt{(-3)^2 - 4\cdot1\cdot1}}{2\cdot1} = \frac{3\pm\sqrt{9 - 4}}{2} = \frac{3\pm\sqrt{5}}{2}.$$