Question

If x = (√5+ √3)/(√5- √3) and y = (√5- √3)/(√5 + √3), then find the value of x2 + y2.

Answer 1

x=(√5+√3)(√5-√3)
  =√5²-√3²                                 (a+b)(a-b)=a²-b²
  =5-3
  =2
y=(√5-√3)(√5+√3)
  =√5²-√3²                                  (a+b)(a-b)=a²-b²
  =5-3
  =2
x²+y²= 2²+2²
        =4+4
        =8


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Answer 2

Answer:

$x^2 + y^2=47+6\sqrt{15}$        

Step-by-step explanation:

Given : $x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$

            $y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$

To Find :$x^2 + y^2$

Solution:

$x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$

$x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} \times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}$

$x=\frac{5+\sqrt{15}+\sqrt{15}+3}{5-3}$

$x=\frac{8+2\sqrt{15}}{2}$

$x=4+\sqrt{15}$

$y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}\times \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}$

$y=\frac{5-\sqrt{15}-\sqrt{15}-3}{5-3}$

$y=\frac{2-2\sqrt{15}}{2}$

$y=1-\sqrt{15}$

$x^2 + y^2$

Substitute the values

$(4+\sqrt{15})^2 + (1-\sqrt{15})^2$

$16+15+8\sqrt{15}+ 1+15-2\sqrt{15}$

$47+6\sqrt{15}$

Hence $x^2 + y^2=47+6\sqrt{15}$