Final answer:
The video game company's profit function is a quadratic equation with a negative discriminant, indicating no real roots. Therefore, the company will not break even as the function never crosses the x-axis, and the correct answer is B) No.
Step-by-step explanation:
In order to determine if the video game company will break even in its first year, we need to find the roots of the profit function, which is P(x) = -0.8x² + 3x - 4.6. The company breaks even when the profit P(x) is equal to zero. We will solve the equation -0.8x² + 3x - 4.6 = 0 to see if there are any real solutions. This is a quadratic equation and can be solved by factoring, completing the square, or using the quadratic formula.
Without explicitly solving the equation, we can use the fact that the profit function is a downwards-opening parabola (because the coefficient of x² is negative). This implies that the vertex of the parabola is a maximum point, and the profit function will have zero, one, or two points where it intersects the x-axis, depending on the discriminant, which in a general form is b² - 4ac.
For the given profit function, the discriminant can be calculated as:
Δ = b² - 4ac = (3)² - 4(-0.8)(-4.6) = 9 - 14.88 = -5.88
Since the discriminant is negative, the equation does not have any real roots, which means that the function does not cross the x-axis, and hence the company will not break even. Therefore, the answer is B) No, the company will not break even.