Question

Știind că
$x - \frac{1}{x} = 2$
, calculați
${x}^{2} + \frac{1}{x {}^{2} }$
​

Answer 1

Răspuns:

Explicație pas cu pas:

( x - 1/x)² = 2²

x² - 2 · x · (1/x) + 1²/x² = 4

x² - 2 + 1/x² = 4

x² + 1/x² = 4 + 2

x² + 1/x² = 6

Answer 2

$x - \dfrac{1}{x} = 2\,\Big|\verb ^2 \\\\ \\\Leftrightarrow\,\,\Big(x-\dfrac{1}{x}\Big)^2 = 2^2 \\ \\ \Leftrightarrow\,\,x^2-2\cdot x\cdot \dfrac{1}{x}+\Big(\dfrac{1}{x}\Big)^2 = 4 \\ \\ \Leftrightarrow\,\, x^2-2\cdot 1+\dfrac{1}{x^2} = 4 \\ \\ \Leftrightarrow\,\,x^2 +\dfrac{1}{x^2} = 4+2 \\ \\ \Leftrightarrow\,\, \boxed{x^2+\dfrac{1}{x^2} = 6}$