(2y)^2+y^2=10000

Solution for (2y)^2+y^2=10000 equation:

(2y)^2+y^2=10000

We move all terms to the left:

(2y)^2+y^2-(10000)=0

We add all the numbers together, and all the variables

3y^2-10000=0

a = 3; b = 0; c = -10000;
Δ = b2-4ac
Δ = 02-4·3·(-10000)
Δ = 120000
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{120000}=\sqrt{40000*3}=\sqrt{40000}*\sqrt{3}=200\sqrt{3}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-200\sqrt{3}}{2*3}=\frac{0-200\sqrt{3}}{6}
=-\frac{200\sqrt{3}}{6}
=-\frac{100\sqrt{3}}{3}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+200\sqrt{3}}{2*3}=\frac{0+200\sqrt{3}}{6}
=\frac{200\sqrt{3}}{6}
=\frac{100\sqrt{3}}{3}
$