Question

sendo A= (x+1/x) ao quadrado e B= (X - 1/x) ao quadrado, calcule (a-b) ao quadrado.

alguém ajuda pfv. Obrigado.​

Answer 1

$A~=~(x+\frac{1}{x})^2\\\\A~=~(x+\frac{1}{x}).(x+\frac{1}{x})\\\\A~=~x.x+x.\frac{1}{x}+\frac{1}{x}.x+\frac{1}{x}.\frac{1}{x}\\\\A~=~x^2+1+1+\frac{1}{x^2}\\\\\boxed{A~=~x^2+2+\frac{1}{x^2}}$

$B~=~(x-\frac{1}{x})^2\\\\B~=~(x-\frac{1}{x}).(x-\frac{1}{x})\\\\B~=~x.x+x.-\frac{1}{x}-\frac{1}{x}.x-\frac{1}{x}.-\frac{1}{x}\\\\B~=~x^2-1-1+\frac{1}{x^2}\\\\\boxed{B~=~x^2-2+\frac{1}{x^2}}$

$(A-B)^2~=~\left[~\left(x^2+2+\frac{1}{x^2}\right)-\left(x^2-2+\frac{1}{x^2}\right)~\right]^2\\\\(A-B)^2~=~\left[~x^2-x^2+2+2+\frac{1}{x^2}-\frac{1}{x^2}~\right]^2\\\\(A-B)^2~=~4^2\\\\\boxed{(A-B)^2~=~16}$