if x^2+1/x^2=17/4, then find 2/5(x+1/x)+(x-1/x)
You asked:
Given \({x}^{2} + \frac{1}{{x}^{2}} = \frac{17}{4}\) evaluate the expression \(\frac{2}{5 \cdot \left(x + \frac{1}{x}\right)} + x - \frac{1}{x}\).
MathBot Answer:
\(x + \frac{2}{5 \left(x + \frac{1}{x}\right)} - \frac{1}{x} \in \left\{- \frac{83}{50}, - \frac{67}{50}, \frac{67}{50}, \frac{83}{50}\right\}\)
The 4 solutions to the equation are: \[\begin{aligned}x &= -2\\x &= - \frac{1}{2} = -0.5\\x &= \frac{1}{2} = 0.5\\x &= 2\end{aligned}\]
By substituting this value into the expression, we find that: $$\begin{aligned}x + \frac{2}{5 \left(x + \frac{1}{x}\right)} - \frac{1}{x} &= -2 + \frac{2}{5 \left(-2 + \frac{1}{-2}\right)} - \frac{1}{-2}\\&= - \frac{83}{50}\end{aligned}$$