Question

if x=√5+1÷√5-1 and y=√5-1÷√5+1 find the value of x^2+ xy +y^2

Answer 1

Answer:

$\bf\;x^2+xy+y^2=8$

Step-by-step explanation:

Given:

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

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

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

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

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

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

$\implies\bf\;x+y=3$

Now

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

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

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

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

$\implies\boxed{\bf\;x^2+xy+y^2=8}$

Answer 2

answer is above plz mark as Brainliest