First, we "play with multipliers of #8# (1x8, 2x4, 4x2 8x1) and 1 to factor and then solve each term for #0#:

#8x^2 - 6x + 1 = 0#

#(4x - 1)(2x - 1) = 0#

Now we solve each term of the factored quadratic for #0#:

Solution 1)

#4x - 1 = 0#

#4x - 1 + color(red)(1) = 0 + color(red)(1)#

#4x - 0 = 1#

#4x = 1#

#(4x)/color(red)(4) = 1/color(red)(4)#

#(color(red)(cancel(color(black)(4)))x)/cancel(color(red)(4)) = 1/4#

#x = 1/4#

Solution 2)

#2x - 1 = 0#

#2x - 1 + color(red)(1) = 0 + color(red)(1)#

#2x - 0 = 1#

#2x = 1#

#(2x)/color(red)(2) = 1/color(red)(2)#

#(color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) = 1/2#

#x = 1/2#