If x = 2-√5/ 2+√5 and y = 2+√5/ 2-√5, find the value of x-y+xy
Answers
Step-by-step explanation:
We know that,
(x + y)(x - y) = x² - y²
Therefore,
\begin{gathered}x^2 - y^2 = (\frac{2 - \sqrt{5}}{2 + \sqrt{5}} + \frac{2 + \sqrt{5}}{2 - \sqrt{5}})(\frac{2 - \sqrt{5}}{2 + \sqrt{5}} - \frac{2 + \sqrt{5}}{2 - \sqrt{5}}) \\ \\ = (\frac{(2 - \sqrt{5})^2 + (2 + \sqrt{5})^2}{(2 + \sqrt{5})(2 - \sqrt{5})})(\frac{(2 - \sqrt{5})^2 - (2 + \sqrt{5})^2}{(2 + \sqrt{5})(2 - \sqrt{5})}) \\ \\ = (\frac{9 - 4\sqrt{5} + 9 + 4\sqrt{5}}{4 - 5})(\frac{9 - 4\sqrt{5} - 9 - 4\sqrt{5}}{4 - 5}) \\ \\ = \frac{18}{- 1} \times \frac{- 8\sqrt{5}}{- 1} = \frac{- 144\sqrt{5}}{1} = - 144\sqrt{5}\end{gathered}
x
2
−y
2
=(
2+
5
2−
5
+
2−
5
2+
5
)(
2+
5
2−
5
−
2−
5
2+
5
)
=(
(2+
5
)(2−
5
)
(2−
5
)
2
+(2+
5
)
2
)(
(2+
5
)(2−
5
)
(2−
5
)
2
−(2+
5
)
2
)
=(
4−5
9−4
5
+9+4
5
)(
4−5
9−4
5
−9−4
5
)
=
−1
18
×
−1
−8
5
=
1
−144
5
=−144
5
- 144√5 is the answer.
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