x^2 + 1/x^2 = 254 then x+1/x=?

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on Apr 10, 2025 at 2:19 am



MathBot Answer:

x+1x∈{−8−37+1−8−37,−37+8+18−37,1−8+37−8+37,137+8+37+8}x + \frac{1}{x} \in \left\{-8 - 3 \sqrt{7} + \frac{1}{-8 - 3 \sqrt{7}}, - 3 \sqrt{7} + 8 + \frac{1}{8 - 3 \sqrt{7}}, \frac{1}{-8 + 3 \sqrt{7}} - 8 + 3 \sqrt{7}, \frac{1}{3 \sqrt{7} + 8} + 3 \sqrt{7} + 8\right\}

The 4 solutions to the equation are: x=−8−37≈−15.937254x=−8+37≈−0.062746067x=8−37≈0.062746067x=8+37≈15.937254\begin{aligned}x &= -8 - 3 \sqrt{7} \approx -15.937254\\x &= -8 + 3 \sqrt{7} \approx -0.062746067\\x &= 8 - 3 \sqrt{7} \approx 0.062746067\\x &= 8 + 3 \sqrt{7} \approx 15.937254\end{aligned}

By substituting this value into the expression, we find that: x+1x=(−8−37)+1−8−37=−8−37+1−8−37\begin{aligned}x + \frac{1}{x} &= \left(-8 - 3 \sqrt{7}\right) + \frac{1}{-8 - 3 \sqrt{7}}\\&= -8 - 3 \sqrt{7} + \frac{1}{-8 - 3 \sqrt{7}}\end{aligned}

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