X+x^2=3x^2-1

Solution for X+x^2=3x^2-1 equation:

X+X^2=3X^2-1

We move all terms to the left:

X+X^2-(3X^2-1)=0

We get rid of parentheses

X^2-3X^2+X+1=0

We add all the numbers together, and all the variables

-2X^2+X+1=0

a = -2; b = 1; c = +1;
Δ = b2-4ac
Δ = 12-4·(-2)·1
Δ = 9
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{9}=3$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-3}{2*-2}=\frac{-4}{-4}
=1
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+3}{2*-2}=\frac{2}{-4}
=-1/2
$