Question

if x equal to root 5 minus root 2 upon root 5 + root 2 and Y is equal to root 5 + root 2 upon root 5 minus root 2 find the value of x square + xy + Y square

Answer 1

$\textbf{Given:}$

$\mathsf{x=\dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}}$

$\mathsf{y=\dfrac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}}$

$\textbf{To find:}$

$\textsf{The value of}\;\mathsf{x^2+xy+y^2}$

$\textbf{Solution:}$

$\mathsf{Consider,}$

$\mathsf{x+y=\dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}+\dfrac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}}$

$\mathsf{x+y=\dfrac{(\sqrt{5}-\sqrt{2})^2+(\sqrt{5}+\sqrt{2})^2}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})}}$

$\mathsf{x+y=\dfrac{5+2-2\sqrt{5}\sqrt{2}+5+2+2\sqrt{5}\sqrt{2}}{5-2}}$

$\mathsf{x+y=\dfrac{14}{3}}$

$\mathsf{Now,}$

$\mathsf{x^2+xy+y^2}$

$\textsf{Add and subtract xy}$

$\mathsf{=(x^2+2xy+y^2)-xy}$

$\mathsf{=(x+y)^2-xy}$

$\mathsf{=\left(\dfrac{14}{3}\right)^2-\dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}{\times}\dfrac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}}$

$\mathsf{=\dfrac{196}{9}-1}$

$\mathsf{=\dfrac{196-9}{9}}$

$\mathsf{=\dfrac{187}{9}}$

$\implies\boxed{\mathsf{x^2+xy+y^2=\dfrac{187}{9}}}$

$\textbf{Find more:}$

If x = √7+ √3 and xy = 4, then x4 + y4 =​

https://brainly.in/question/16537214

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

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Answer 2

Step-by-step explanation:

Hope it helps uh!!!!!!!