Question

. If x = 2√5 + √3 and y = 2√5 - √3, evaluate x2 + y2
whole explanation​

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:x^2\:+\:y^2\:=\:46\:}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:x\:=\:2\:\sqrt{5}\:+\:\sqrt{3}\:\&}$

$\displaystyle{\sf\:y\:=\:2\:\sqrt{5}\:-\:\sqrt{3}}$

We have to find the value of x² + y².

Now, we know that,

$\displaystyle{\pink{\sf\:x^2\:+\:y^2\:=\:(\:x\:+\:y\:)^2\:-\:2\:xy}}$

$\displaystyle{\implies\sf\:x^2\:+\:y^2\:=\:(\:2\:\sqrt{5}\:+\:\cancel{\sqrt{3}}\:+\:2\:\sqrt{5}\:-\:\cancel{\sqrt{3}}\:)^2\:-\:2\:\times\:(\:2\:\sqrt{5}\:+\:\sqrt{3}\:)\:(\:2\:\sqrt{5}\:-\:\sqrt{3}\:)}$

$\displaystyle{\implies\sf\:x^2\:+\:y^2\:=\:(\:4\:\sqrt{5}\:)^2\:-\:2\:\times\:[\:(\:2\:\sqrt{5}\:)^2\:-\:(\:\sqrt{3}\:)^2\:]\:\quad\:\dots\:[\:(\:a\:+\:b\:)\:(\:a\:-\:b\:)\:=\:a^2\:-\:b^2\:]}$

$\displaystyle{\implies\sf\:x^2\:+\:y^2\:=\:16\:\times\:5\:-\:2\:\times\:[\:(\:4\:\times\:5\:)\:-\:(\:3\:)\:]}$

$\displaystyle{\implies\sf\:x^2\:+\:y^2\:=\:80\:-\:2\:\times\:(\:20\:-\:3\:)}$

$\displaystyle{\implies\sf\:x^2\:+\:y^2\:=\:80\:-\:2\:\times\:17}$

$\displaystyle{\implies\sf\:x^2\:+\:y^2\:=\:80\:-\:34}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:x^2\:+\:y^2\:=\:46\:}}}}$

Answer 2

x²+y³= 46 is the correct answer....