source: #youtube https://www.youtube.com/watch?v=R0juw4faGck tags: #algebraic-manipulation #do-not-find-x course: [[Algebra 2]] status: #solved ### Problem Given $x+\frac{1}{x}=7$. Find $\frac{x^{2}}{x^{4}+x^{2}+1}$ #### Solution Find a useful expression by squaring the given equation. $\begin{aligned} x+\frac{1}{x}=7\implies \left(x+\frac{1}{x}\right)^2&=7^2\\ x^2+2+\frac{1}{x^2}&=49\\ x^2+1+\frac{1}{x^2}&=48 \quad\star \end{aligned}$ Then use it in the desired expression $\begin{aligned} \frac{x^{2}}{x^{4}+x^{2}+1}&=\\ \frac{1}{x^{2}+1+\frac{1}{x^2}}&=\boxed{\frac{1}{48}} \end{aligned}$