Question

Sendo x²+1/x²=5, calcule o valor de: (x+1/x)² e também de (x-1/x) ²

Answer 1

a) $\left(x+\dfrac{1}{x}\right)^2=x^2+2\cdot x\cdot\dfrac{1}{x}+\dfrac{1}{x^2}$

$\left(x+\dfrac{1}{x}\right)^2=x^2+2+\dfrac{1}{x^2}$

Como $x^2+\dfrac{1}{x^2}=5$, segue que:

$\left(x+\dfrac{1}{x}\right)^2=5+2$

$\left(x+\dfrac{1}{x}\right)^2=7$

b) $\left(x-\dfrac{1}{x}\right)^2=x^2-2\cdot x\cdot\dfrac{1}{x}+\dfrac{1}{x^2}$

$\left(x-\dfrac{1}{x}\right)^2=x^2-2+\dfrac{1}{x^2}$

$\left(x-\dfrac{1}{x}\right)^2=5-2$

$\left(x-\dfrac{1}{x}\right)^2=3$