Question

if x= (√5+1)/(√5-1)And y=(√5-1)/(√5+1),find the value of x2+xy+y2
​

Answer 1

Answer:

x

2

+xy+y

2

=8

Step-by-step explanation:

Given:

x=\frac{\sqrt{5}+1}{\sqrt{5}-1}x=

5

−1

5

+1

y=\frac{\sqrt{5}-1}{\sqrt{5}+1}y=

5

+1

5

−1

x+y=\frac{\sqrt{5}+1}{\sqrt{5}-1}+\frac{\sqrt{5}-1}{\sqrt{5}+1}x+y=

5

−1

5

+1

+

5

+1

5

−1

x+y=\frac{(\sqrt{5}+1)^2+(\sqrt{5}-1)^2}{(\sqrt{5}-1)(\sqrt{5}+1)}x+y=

(

5

−1)(

5

+1)

(

5

+1)

2

+(

5

−1)

2

x+y=\frac{5+1+2\sqrt{5}+5+1-2\sqrt{5}}{\sqrt{5}^2,-1^2}x+y=

5

2

,−1

2

5+1+2

5

+5+1−2

5

x+y=\frac{12}{4}x+y=

4

12

\implies\bf\;x+y=3⟹x+y=3

Now

x^2+xy+y^2=(x^2+2xy+y^2)-xyx

2

+xy+y

2

=(x

2

+2xy+y

2

)−xy

x^2+xy+y^2=(x+y)^2-xyx

2

+xy+y

2

=(x+y)

2

−xy

x^2+xy+y^2=3^2-(\frac{\sqrt{5}+1}{\sqrt{5}-1}{\times}\frac{\sqrt{5}-1}{\sqrt{5}+1})x

2

+xy+y

2

=3

2

−(

5

−1

5

+1

×

5

+1

5

−1

)

x^2+xy+y^2=9-1x

2

+xy+y

2

=9−1

\implies\boxed{\bf\;x^2+xy+y^2=8}⟹

x

2

+xy+y

2

=8