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