Question

REPEDE VA ROG DAU COROANAA!!! Calculati: (1/ radical din 5 +1)² + (1/ radical din 5 -1)²- 3 supra 5
(+1 si -1 nu sunt sub fractie)

Answer 1

$( \frac{1}{ \sqrt{5} } + 1)^{2} + ( \frac{1}{ \sqrt{5} } - 1)^{2} - \frac{3}{5}$
Rationalizezi numitorii:
$\frac{1}{\sqrt{5} } = \frac{ \sqrt{5} }{ \sqrt{5} \times \sqrt{5} } = \frac{ \sqrt{5} }{5}$
=>
${( \frac{ \sqrt{5} }{5} + 1)}^{2} + {( \frac{ \sqrt{5} }{5} - 1)}^{2} - \frac{3}{5}$
Folosesti formula binomului:
$(x + y)^{2} = {x}^{2} + 2xy + {y}^{2} \\ (x - y)^{2} = {x}^{2} - 2xy + {y}^{2}$
Desfaci parantezele:
${( \frac{ \sqrt{5} }{5} + 1) }^{2} = {( \frac{ \sqrt{5} }{5} )}^{2} + 2 \times \frac{ \sqrt{5} }{5} \times 1 + {1}^{2} \\ = \frac{5}{25} + \frac{ 2\sqrt{5} }{5} + 1$

${( \frac{ \sqrt{5} }{5} - 1) }^{2} = \frac{5}{25} - \frac{ \sqrt{5} }{5} + 1$
Revenim la expresie:

$\frac{5}{25} + \frac{2 \sqrt{5} }{5} + 1 + \frac{5}{25} - \frac{2 \sqrt{5} }{5} + 1 - \frac{3}{5} \\ \frac{10}{25} - \frac{3}{5} + 2 \\ \frac{10}{25} - \frac{15}{25} + 2 \\ - \frac{5}{25} + 2 \\ - \frac{1}{5} + 2 \\ - \frac{1}{5} + \frac{10}{5} \\ \frac{9}{5}$