Final answer:
The quadratic equation 3x^2 + 2 = 0 has two complex solutions, x = ±(sqrt(2/3))i. Neither of the student's proposed solutions is correct. The correct solutions involve the square root of 2/3 multiplied by the imaginary unit i.
Step-by-step explanation:
The student's question pertains to finding the solutions to the quadratic equation 3x2 + 2 = 0. To solve this equation, we must use the square root method since the equation is already set with the x2 term isolated and doesn't have an x term (meaning b = 0 in the general form of ax2 + bx + c = 0).
First, we isolate x2 as follows:
- Subtract 2 from both sides of the equation: 3x2 = -2.
- Divide both sides of the equation by 3 to get x2 = -2/3.
- Take the square root of both sides. Since we are dealing with a negative number, we introduce i, the imaginary unit, such that i2 = -1, to get x = ±sqrt(-2/3).
- Simplify the square root to get x = ±(sqrt(2/3))i.
We obtained two complex solutions: x = (sqrt(2/3))i and x = -(sqrt(2/3))i, since the square root of a negative number results in an imaginary number. Thus, neither of the student's proposed solutions ((√6/3)i, 2/3, or (2/3)i) is correct. Instead, we discovered that the correct solutions involve the square root of 2/3 multiplied by i.