Question

if x = √ 3+ √2 divides √3 - √2 and y = √3 -√2 divides √3 + √2 find x^2+ y^2​

Answer 1

Question

$\sf{x=\dfrac{\sqrt{3} +\sqrt{2} }{\sqrt{3} -\sqrt{2} } }$ and $\sf{y=\dfrac{\sqrt{3} -\sqrt{2} }{\sqrt{3} +\sqrt{2} } }$ then find x²+y²

Answer

98

Idea

• Algebraic Identity

This equation is always true.

For any number you put in, this is always true.

Before we solve the question

If we look at the fractions

they are reciprocal of one another.

∴$\sf{xy=1}$ ...(1)

After rationalization

$\sf{x=\dfrac{(\sqrt{3} +\sqrt{2} )^2}{1} }$ and $\sf{y=\dfrac{(\sqrt{3} -\sqrt{2} )^2}{1} }$

After calculation we obtain

$\sf{x=5+2\sqrt{6} }$ and $\sf{y=5-2\sqrt{6} }$

∴$\sf{x+y=10}$ ...(2)

Let's solve the problem

$\sf{(x+y)^2=x^2+2xy+y^2}$ ...(3)

From (1), (2), and (3)

$\sf{10^2=x^2+2\times1+y^2}$

$\sf{100-2=x^2+y^2}$

∴$\sf{x^2+y^2=98}$