x^2=12^2+0.25x^2

Solution for x^2=12^2+0.25x^2 equation:

x^2=12^2+0.25x^2

We move all terms to the left:

x^2-(12^2+0.25x^2)=0

We get rid of parentheses

x^2-0.25x^2-12^2=0

We add all the numbers together, and all the variables

0.75x^2-144=0

a = 0.75; b = 0; c = -144;
Δ = b2-4ac
Δ = 02-4·0.75·(-144)
Δ = 432
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:

$\sqrt{\Delta}=\sqrt{432}=\sqrt{144*3}=\sqrt{144}*\sqrt{3}=12\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-12\sqrt{3}}{2*0.75}=\frac{0-12\sqrt{3}}{1.5}
=-\frac{12\sqrt{3}}{1.5}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+12\sqrt{3}}{2*0.75}=\frac{0+12\sqrt{3}}{1.5}
=\frac{12\sqrt{3}}{1.5}
$