Final answer:
The discriminant of the quadratic equation 3x^2 - 4x + 3 = 0 is -20, indicating no real solutions; instead, the equation has two complex solutions: 2/3 + √(5)/3i and 2/3 - √(5)/3i.
Step-by-step explanation:
To find the discriminant and exact solutions of the quadratic equation 3x^2 - 4x + 3 = 0, we use the quadratic formula, which is x = (-b ± √(b^2 - 4ac))/(2a), where a, b, and c are coefficients from the quadratic equation ax^2 + bx + c = 0.
The discriminant is the part of the quadratic formula under the square root, given by Δ = b^2 - 4ac. For our equation, a = 3, b = -4, and c = 3, so the discriminant is Δ = (-4)^2 - 4(3)(3) = 16 - 36 = -20. Since the discriminant is negative, we know that there are no real solutions to this equation, and the solutions are complex numbers.
The exact solutions, given the discriminant, are x = (4 ± √-20)/6. We can express this as x = 2/3 ± √(5)/3i, using the imaginary unit i, where i^2 = -1.