Question

Simplificati raportul:
a) $\frac{ x^{2}-5 }{ x^{}+ \sqrt{5} }$
b) $\frac{ \sqrt{2}-2 }{ \sqrt{2}-1 }$

Answer 1

$a)\frac{x^{2}-5 }{x+ \sqrt{5} } = \frac{(x- \sqrt{5} )(x^{2}-5) }{x+ \sqrt{5} } \\ \\ = \frac{x^{3}-5x- x^{2} \sqrt{5} +5 \sqrt{5} }{x+ \sqrt{5}} \\ \\ = \frac{ x^{2} ( \sqrt{5} +x)-5( \sqrt{5} +x)}{x+ \sqrt{5} } \\ \\ = \frac{x+ \sqrt{5} ( x^{2} -5)}{x+ \sqrt{5} } \\ \\ = x^{2} -5$
 $b)\frac{ \sqrt{2} -2}{ \sqrt{2}-1 } = \frac{ (\sqrt{2} -2)( \sqrt{2}-1) }{({ \sqrt{2}-1 })({ \sqrt{2}+1 })} \\ \\ \frac{2-2 \sqrt{2}+ \sqrt{2} -2}{2+ \sqrt{2}- \sqrt{2} -1 } \\ \\ =- \sqrt{2}$