Question

If X=5+3√5/5-3√5 and y= 5-3√5/5+3√5 find value of x²+y²

Answer 1

Answer:

Step-by-step explanation:

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

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

$x=\frac{5+\sqrt{15}+\sqrt{15}+3}{5-3} \\\\x=\frac{8+2\sqrt{15}}{2} \\x=4+\sqrt{15}\\\\y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}\times \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}\\y=\frac{5-\sqrt{15}-\sqrt{15}-3}{5-3}\\\\y=\frac{2-2\sqrt{15}}{2}\\y=1-\sqrt{15}x^2 + y^2$                        

$\\x^2 + y^2\\\\Substitute the values\\\\(4+\sqrt{15})^2 + (1-\sqrt{15})^2\\\\= 16+15+8\sqrt{15}+ 1+15-2\sqrt{15}47+6\sqrt{15}\\\\Hence x^2 + y^2\\\\=47+6\sqrt{15}$