x^2+2x^2=500^2

Solution for x^2+2x^2=500^2 equation:

x^2+2x^2=500^2

We move all terms to the left:

x^2+2x^2-(500^2)=0

We add all the numbers together, and all the variables

3x^2-250000=0

a = 3; b = 0; c = -250000;
Δ = b2-4ac
Δ = 02-4·3·(-250000)
Δ = 3000000
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{3000000}=\sqrt{1000000*3}=\sqrt{1000000}*\sqrt{3}=1000\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-1000\sqrt{3}}{2*3}=\frac{0-1000\sqrt{3}}{6}
=-\frac{1000\sqrt{3}}{6}
=-\frac{500\sqrt{3}}{3}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+1000\sqrt{3}}{2*3}=\frac{0+1000\sqrt{3}}{6}
=\frac{1000\sqrt{3}}{6}
=\frac{500\sqrt{3}}{3}
$