6x=2x2+1

Solution for 6x=2x2+1 equation:

6x=2x^2+1

We move all terms to the left:

6x-(2x^2+1)=0

We get rid of parentheses

-2x^2+6x-1=0

a = -2; b = 6; c = -1;
Δ = b2-4ac
Δ = 62-4·(-2)·(-1)
Δ = 28
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:

$\sqrt{\Delta}=\sqrt{28}=\sqrt{4*7}=\sqrt{4}*\sqrt{7}=2\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{7}}{2*-2}=\frac{-6-2\sqrt{7}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{7}}{2*-2}=\frac{-6+2\sqrt{7}}{-4}
$