Math, asked by adityasaini9412, 7 months ago

If x = 2-√5/ 2+√5 and y = 2+√5/ 2-√5, find the value of x-y+xy​

Answers

Answered by chakmajhimit1
0

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.

Hope this may be helpful.

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