Question

Without solving, determine the character of the solutions of the equation in the complex number system.

3x²-3x+1=0
Choose the sentence that describes the character of the solutions to the quadratic equation
A.The equation has a repeated real solution.
B.The equation has two unequal real solutions.
c. the equation has two complex solutions

Answer 1

The quadratic equation 3x²-3x+1=0 has a negative discriminant, which indicates that its solutions are two complex numbers. The correct answer is option C: the equation has two complex solutions.

Step-by-step explanation:

• To determine the character of the solutions of the quadratic equation 3x²-3x+1=0, we use the discriminant method.
• The discriminant (D) is given by D = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation ax²+bx+c = 0. For our equation, a=3, b=-3, and c=1.
• Thus, the discriminant is D = (-3)² - 4(3)(1) = 9 - 12 = -3.
• Since the discriminant is negative (D < 0), this means the quadratic equation has two complex solutions. If the discriminant had been zero, the equation would have a repeated real solution, and if the discriminant had been positive, it would have two unequal real solutions.
• Therefore, in this case, the correct answer is C: the equation has two complex solutions.