Mathematico | Mathematico

The roots of the equation

x + 4 = \frac{k}{x}

are $α$ and $β$ .

Given that $α = \sqrt{5} - 2$ , find the value of $α^{2} + β^{2} + k^{2}$ .

Hint

It's best to multiply by $x$ to get

x^{2} + 4 x - k = 0

Now substitute $x = \sqrt{5} - 2$ to find the value of $k .$

Hint

Once you know $k,$ this becomes a quadratic which you can solve using the formula.

Solution

Let's simplify first

\begin{aligned} x + 4 & = \frac{k}{x} \\ x^{2} + 4 x & = k \\ x^{2} + 4 x - k & = 0 \end{aligned}

We know that $x = \sqrt{5} - 2$ solves the equation, so

\begin{aligned} (\sqrt{5} - 2)^{2} - 4 (\sqrt{5} - 2) & = k \\ 9 - 4 \sqrt{5} + 4 \sqrt{5} - 8 & = k \\ k & = 1 \end{aligned}

Our original equation is therefore

\begin{aligned} x^{2} + 4 x - 1 & = 0 \\ x & = \frac{- 4 \pm \sqrt{16 - 4 (1) (- 1)}}{2} \\ = - 2 \pm \sqrt{5} \end{aligned}

So $α^{2} + β^{2} + k^{2} = 19$