32x^2=16x+1

Solution for 32x^2=16x+1 equation:

32x^2=16x+1

We move all terms to the left:

32x^2-(16x+1)=0

We get rid of parentheses

32x^2-16x-1=0

a = 32; b = -16; c = -1;
Δ = b2-4ac
Δ = -162-4·32·(-1)
Δ = 384
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:

$\sqrt{\Delta}=\sqrt{384}=\sqrt{64*6}=\sqrt{64}*\sqrt{6}=8\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-16)-8\sqrt{6}}{2*32}=\frac{16-8\sqrt{6}}{64}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-16)+8\sqrt{6}}{2*32}=\frac{16+8\sqrt{6}}{64}
$