$$ \begin{aligned} x^2+\frac{1}{x^2} &= 66&& \text{multiply ALL terms by } \color{blue}{ x^2 }. \\[1 em]x^2x^2+x^2\cdot\frac{1}{x^2} &= x^2\cdot66&& \text{cancel out the denominators} \\[1 em]x^4+1 &= 66x^2&& \text{move all terms to the left hand side } \\[1 em]x^4+1-66x^2 &= 0&& \text{simplify left side} \\[1 em]x^4-66x^2+1 &= 0&& \\[1 em] \end{aligned} $$

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