x^2+x^2=1000

Solution for x^2+x^2=1000 equation:

x^2+x^2=1000

We move all terms to the left:

x^2+x^2-(1000)=0

We add all the numbers together, and all the variables

2x^2-1000=0

a = 2; b = 0; c = -1000;
Δ = b2-4ac
Δ = 02-4·2·(-1000)
Δ = 8000
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:

$\sqrt{\Delta}=\sqrt{8000}=\sqrt{1600*5}=\sqrt{1600}*\sqrt{5}=40\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-40\sqrt{5}}{2*2}=\frac{0-40\sqrt{5}}{4}
=-\frac{40\sqrt{5}}{4}
=-10\sqrt{5}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+40\sqrt{5}}{2*2}=\frac{0+40\sqrt{5}}{4}
=\frac{40\sqrt{5}}{4}
=10\sqrt{5}
$