Question

x - (1/x) = √2. atunci x³ - (1/x³) = ?

Answer 1

$x- \frac{1}{x} = \sqrt{2}$
Ridicam la patrat in ambii membri.
$x^{2} + \frac{1}{ x^{2} } - \frac{2x}{x} =2$
$x^{2} + \frac{1}{ x^{2} } - 2 =2$
$x^{2} + \frac{1}{ x^{2} } =4$
Inmultim in ambii membri cu prima relatie
$(x^{2} + \frac{1}{ x^{2} })(x- \frac{1}{x}) =4 \sqrt{2}$
$x^{3}- \frac{1}{ x^{3} }- x + \frac{1}{ x} =8$
$x^{3} - \frac{1}{ x^{3} } - \frac{ x^{2} }{x} + \frac{x}{ x^{2} } =8$
$x^{3}- \frac{1}{ x^{3} }+ \sqrt{2} =8$
$x^{3}- \frac{1}{ x^{3} }=8- \sqrt{2}$