Mathematico | Mathematico

The roots of the equation

\frac{1}{x^{2}} + \frac{3}{x} = 1

are $α$ and $β$ .

Find the value of

α^{2} + β^{2}

Hint

Multiply both sides of the equation by $x^{2}$ to begin with.

Solution

\begin{aligned} \frac{1}{x^{2}} + \frac{3}{x} & = 1 \\ 1 + 3 x & = x^{2} \\ x^{2} - 3 x - 1 & = 0 \\ {(x - \frac{3}{2})}^{2} - \frac{9}{4} - 1 & = 0 \\ {(x - \frac{3}{2})}^{2} & = \frac{13}{4} \\ x & = \frac{3}{2} \pm \sqrt{\frac{13}{4}} \\ = \frac{3 \pm \sqrt{13}}{2} \end{aligned}