Answer:
x = (3 + √-3) / 6 and x = (3 - √-3) / 6, so answers A and B are correct.
Step-by-step explanation:
To clarify, when we say "roots" of a quadratic expression, we're talking about it's x-intercepts, or where the entire expression = 0. Let's start with that:
3x² - 3x + 1 = 0
Let's complete the square to solve it:
9x² - 9x + 3 = 0
9x² - 9x + 2.25 = -0.75
(3x - 1.5)² = -0.75
(3x - 3 / 2)² = -3 / 4
3x - 3 / 2 = ± i√(3 / 4)
3x - 3 / 2 = ± i√3 / 2
3x = 3 / 2 ± i√3 / 2
3x = (3 ± i√3) / 2
x = (3 ± i√3) / 6
So the x is equal to both (3 + i√3) / 6 and (3 - i√3) / 6.
You'll note that these answers don't exactly match the given ones. The only difference there is that I factored -1 out of the radical, giving you i as a coefficient to it. You may not yet be at a point where you've covered imaginary numbers, but I assure you the answers I found here are exactly the same as x = (3 + √-3) / 6 and x = (3 - √-3) / 6.