Question

Sabendo-se que x+1/x = 3√2/2, então x²+(1/x)² é igual a:

a)3/2
b)5/2
c)3
d)7/2
e)9/2

Answer 1

É dado que

    $\mathsf{x+\dfrac{1}{x}=\dfrac{3\sqrt{2}}{2}}$

Eleve os dois lados ao quadrado:

    $\Big(\mathsf{x+\dfrac{1}{x}\Big)^{\!2}=\Big(\dfrac{3\sqrt{2}}{2}\Big)^{\!2}}\\\\\\\\\Big(\mathsf{x+\dfrac{1}{x}\Big)^{\!2}=\dfrac{(3\sqrt{2})^2}{2^2}}\\\\\\\\\Big(\mathsf{x+\dfrac{1}{x}\Big)^{\!2}=\dfrac{9\cdot 2}{2^2}}\\\\\\\Big(\mathsf{x+\dfrac{1}{x}\Big)^{\!2}=\dfrac{9}{2}}$

Expanda o quadrado da soma no lado esquerdo:

    $\mathsf{x^2+2\cdot \diagup\!\!\!\! x\cdot \dfrac{1}{\diagup\!\!\!\! x}+\Big(\dfrac{1}{x}\Big)^{\!2}=\dfrac{9}{2}}\\\\\\\mathsf{x^2+2+\dfrac{1}{x^2}=\dfrac{9}{2}}\\\\\\\mathsf{x^2+\dfrac{1}{x^2}=\dfrac{9}{2}-2}\\\\\\\mathsf{x^2+\dfrac{1}{x^2}=\dfrac{9}{2}-\dfrac{4}{2}}$

    $\mathsf{x^2+\dfrac{1}{x^2}=\dfrac{5}{2}\quad\longleftarrow\quad resposta:~alternativa~b).}$

Bons estudos! :-)