Is y² + x√(4x² + y²) = y² + (x/y)√(4(x/y)² + 1) for positive x and y?

[cbarker1]

Hello,

I need some aid to see if this true:

$y^2+x*\sqrt{4x^2+y^2}=y^2+\frac{x}{y}*\sqrt{4(\frac{x}{y})^2+1}$ provide that y>0 and x>0.Thank you,

Cbarker1

 

[evinda]

Hello,

I need some aid to see if this true:

$y^2+x*\sqrt{4x^2+y^2}=y^2+\frac{x}{y}*\sqrt{4(\frac{x}{y})^2+1}$ provide that y>0 and x>0.Thank you,

Cbarker1

(Wave)

No. It holds that $y^2+x \cdot \sqrt{4x^2+y^2}=y^2+ x \cdot \sqrt{y^2 \left( \frac{4x^2}{y^2}+1\right)}=y^2+x \cdot |y| \cdot \sqrt{ \frac{4x^2}{y^2}+1}$.And provided that $y>0$, $y^2+x \cdot |y| \cdot \sqrt{ \frac{4x^2}{y^2}+1}=y^2+x \cdot y \cdot \sqrt{ \frac{4x^2}{y^2}+1}$