Question

Sabendo que √x + 1/√x = 3, determine o valor de (x - 1/x)²

Answer 1

$\displaystyle \sf temos: \ \sqrt{x} +\frac{1}{\sqrt{x}} = 3 \\\\\\ queremos : \left(x-\frac{1}{x}\right)^2\ \\\\\\ \left(x-\frac{1}{x}\right)^2 =x^2-2\cdot x\cdot \frac{1}{x^2}+\frac{1}{x^2} \\\\\\ \left( x-\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2} - 2$

Utilizando a informação que temos :

$\displaystyle \sf \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^2 = 3^2 \\\\\\ x+2\cdot \sqrt{x}\cdot \frac{1}{\sqrt{x}} + \frac{1}{x} = 9 \\\\\\ x +2+\frac{1}{x} = 9 \\\\\\ x+\frac{1}{x} = 7 \to \left(x+\frac{1}{x}\right)^2=7^2 \\\\\\ x^2+2\cdot x\cdot \frac{1}{x}+\frac{1}{x^2} = 49 \\\\\\ x^2+\frac{1}{x^2} = 49-2 \\\\\\ x^2+\frac{1}{x^2} = 47$

Daí :

$\displaystyle \sf \left( x-\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2} - 2 \\\\\\ \left( x-\frac{1}{x}\right)^2 = 47 - 2 \\\\\\ \boxed{\sf \left( x-\frac{1}{x}\right)^2 = 45 \ }\checkmark$