Question

x=(√5+1)/(√5-1) y=(√5-1)/(√5+1) what is x^2+xy+y^2

Answer 1

Is this answer right?

Answer 2

The value of (x+y)²-xy = 8.

• Given that x =(√5 +1)/(√5-1) and y=(√5-1)/(√5+1)
• We have to find the value of x²+xy +y²

We simplify the equation we get,

x²+xy +y² = x²+xy +y² +xy - xy = x²+2xy +y² - xy = (x+y)²-xy

• We have to find the value of (x+y)²-xy
• Now we evaluate the value of (x+y)² and xy

xy = $(\frac{\sqrt{5}+1}{\sqrt{5}-1} )(\frac{\sqrt{5}-1}{\sqrt{5}+1} )$

xy = 1

• Now, (x+y)² =

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

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

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

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

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

= $(\frac{12}{4})^2$

= 3² = 9

• Now, we evaluate (x+y)²-xy by substituting the values

(x+y)²-xy = 9 - 1 = 8