Question

Dado x - 1/x = 13, determine o valor de x² - 1/x²

Answer 1

$\displaystyle \text x - \frac{1}{\text x}=13 \ \ ; \ \ \text x^2-\frac{1}{\text x^2} = \ ? \\\\\\ \text {Sabemos que}: \\\\ \text x^2-\frac{1}{\text x^2}=(\text x-\frac{1}{\text x})(\text x+\frac{1}{\text x}) \\\\\\ \text x^2-\frac{1}{\text x^2}=13(\text x+\frac{1}{\text x}) \\\\ \text{Elevando ao quadrado a express{\~a}o que temos} : \\\\ (\text x-\frac{1}{\text x})^2 = \text x^2+\frac{1}{\text x^2}-\frac{2\text x}{\text x} \\\\\\ 13^2+2=\text x^2+\frac{1}{\text x^2}$

$\displaystyle \text x^2+\frac{1}{\text x ^2} = 171 \\\\ \text{Fa{\c c}amos}: \\\\ (\text x+\frac{1}{\text x})^2=\text x^2+\frac{1}{\text x^2}+2 \\\\\\ (\text x+\frac{1}{\text x})^2=171+2 \\\\\\ \text x+\frac{1}{\text x}=\sqrt{173}$

Portanto :

$\displaystyle \text x^2-\frac{1}{\text x^2}=(\text x-\frac{1}{\text x})(\text x+\frac{1}{\text x}) \\\\\\ \huge\boxed{\text x^2-\frac{1}{\text x^2}=13. \sqrt{173}\ }\checkmark$

OU

$\huge\boxed{\text x^2 -\frac{1}{\text x^2}\approx 170,98\ }\checkmark$