1/x²+x²=7 then find x³-1/x³

asked by guest
on Jan 10, 2025 at 3:54 am



You asked:

Given \(\frac{1}{{x}^{2}} + {x}^{2} = 7\) evaluate the expression \({x}^{3} - \frac{1}{{x}^{3}}\).

MathBot Answer:

\(x^{3} - \frac{1}{x^{3}} \in \left\{\left(- \frac{3}{2} - \frac{\sqrt{5}}{2}\right)^{3} - \frac{1}{\left(- \frac{3}{2} - \frac{\sqrt{5}}{2}\right)^{3}}, \left(- \frac{3}{2} + \frac{\sqrt{5}}{2}\right)^{3} - \frac{1}{\left(- \frac{3}{2} + \frac{\sqrt{5}}{2}\right)^{3}}, - \frac{1}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)^{3}} + \left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right)^{3}, - \frac{1}{\left(\frac{\sqrt{5}}{2} + \frac{3}{2}\right)^{3}} + \left(\frac{\sqrt{5}}{2} + \frac{3}{2}\right)^{3}\right\}\)

The 4 solutions to the equation are: \[\begin{aligned}x &= - \frac{3}{2} - \frac{\sqrt{5}}{2} \approx -2.618034\\x &= - \frac{3}{2} + \frac{\sqrt{5}}{2} \approx -0.38196601\\x &= \frac{3}{2} - \frac{\sqrt{5}}{2} \approx 0.38196601\\x &= \frac{3}{2} + \frac{\sqrt{5}}{2} \approx 2.618034\end{aligned}\]

By substituting this value into the expression, we find that: $$\begin{aligned}x^{3} - \frac{1}{x^{3}} &= \left(- \frac{3}{2} - \frac{\sqrt{5}}{2}\right)^{3} - \frac{1}{\left(- \frac{3}{2} - \frac{\sqrt{5}}{2}\right)^{3}}\\&= \left(- \frac{3}{2} - \frac{\sqrt{5}}{2}\right)^{3} - \frac{1}{\left(- \frac{3}{2} - \frac{\sqrt{5}}{2}\right)^{3}}\end{aligned}$$

+ Submit Question