+26367 if x+ 1/x= 2, what is x2 + 1/x2
if \(x+ \frac{1}{x}= 2\) , what is \( x^2 + \frac{1}{x^2}\)
\(\begin{array}{|rcll|} \hline x+ \frac{1}{x} &=& 2 \quad & | \quad \text{square both sides} \\ \left(~x+ \frac{1}{x}~\right)^2 &=& 2^2 \\ \left(~x+ \frac{1}{x}~\right)^2 &=& 4 \\ x^2 +2\cdot x\cdot \frac{1}{x} + \frac{1}{x^2} &=& 4 \\ x^2 +2 + \frac{1}{x^2} &=& 4 \quad & | \quad -2 \\ x^2 + \frac{1}{x^2} &=& 4-2 \\ \mathbf{x^2 + \frac{1}{x^2}} & \mathbf{=} & \mathbf{2} \\ \hline \end{array} \)
+9466 \(x + \frac{1}{x} = 2 \\ x + \frac{1}{x} - 2 = 0 \\ \frac{x^2}{x} + \frac{1}{x} - \frac{2x}{x} = 0 \\ \frac{x^2+1-2x}{x} = 0\)
We want to know what makes the numerator = 0.
You can also just say multiply both sides by x.
\(x^2+1-2x = 0 \\ (x-1)(x-1) = 0 \\ x = 1\)
(You can easily test this and see that 1 + 1/1 = 2)
So
\(1^2 + \frac{1}{1^2} = 1 + 1 = 2\)
+26367 if x+ 1/x= 2, what is x2 + 1/x2
if \(x+ \frac{1}{x}= 2\) , what is \( x^2 + \frac{1}{x^2}\)
\(\begin{array}{|rcll|} \hline x+ \frac{1}{x} &=& 2 \quad & | \quad \text{square both sides} \\ \left(~x+ \frac{1}{x}~\right)^2 &=& 2^2 \\ \left(~x+ \frac{1}{x}~\right)^2 &=& 4 \\ x^2 +2\cdot x\cdot \frac{1}{x} + \frac{1}{x^2} &=& 4 \\ x^2 +2 + \frac{1}{x^2} &=& 4 \quad & | \quad -2 \\ x^2 + \frac{1}{x^2} &=& 4-2 \\ \mathbf{x^2 + \frac{1}{x^2}} & \mathbf{=} & \mathbf{2} \\ \hline \end{array} \)
