if Y is equal to root 5 + 2 upon root 5 - 2 then find the value of Y square
Answers
=====================
\begin{lgathered}x = \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} + \sqrt{2} } \\ \\ y = \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} } \\ \\ x + y = \frac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} + \sqrt{2} } + \frac{ \sqrt{5} + \sqrt{2} }{ \sqrt{5} - \sqrt{2} } \\ = \frac{ {( \sqrt{5} - \sqrt{2}) }^{2} + {( \sqrt{5} + \sqrt{2} ) }^{2} }{( \sqrt{5} - \sqrt{2})( \sqrt{5} + \sqrt{2} ) } \\ = \frac{5 + 2 - 2 \sqrt{10} + 5 + 2 + 2 \sqrt{10} }{ {( \sqrt{5} )}^{2} - {( \sqrt{2}) }^{2} } \\ = \frac{14}{5 - 2} = \frac{14}{3} \\ \\ xy = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \\ = 1\end{lgathered}
x=
5
+
2
5
−
2
y=
5
−
2
5
+
2
x+y=
5
+
2
5
−
2
+
5
−
2
5
+
2
=
(
5
−
2
)(
5
+
2
)
(
5
−
2
)
2
+(
5
+
2
)
2
=
(
5
)
2
−(
2
)
2
5+2−2
10
+5+2+2
10
=
5−2
14
=
3
14
xy=
5
−
3
5
+
3
×
5
+
3
5
−
3
=1
Now..
x² + xy + y²
= x² + 2xy + y² - xy
= (x +y) ² - xy
\begin{lgathered}= {( \frac{14}{3}) }^{2} - 1 \\ = \frac{196}{9} - 1 \\ = \frac{196 - 9}{9} = \frac{187}{9}\end{lgathered}
=(
3
14
)
2
−1
=
9
196
−1
=
9
196−9
=
9
187
Answer:
21
Step-by-step explanation:
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