Determine the number and type of solutions for the following polynomial function. Make sure to include an explanation in your work to justify how you determined your answer.
P (2) = - 2x1 + 2x2 + 4
The given polynomial function is P(x) = -2x^2 + 2x + 4.
To determine the number and type of solutions for this polynomial function, we can analyze the discriminant of the quadratic equation using the formula:
Discriminant = b^2 - 4ac
Where a = -2, b = 2, and c = 4.
Plugging these values into the formula, we get:
Discriminant = (2)^2 - 4(-2)(4)
Discriminant = 4 + 32
Discriminant = 36
Since the discriminant is positive (36 > 0), the quadratic equation has two distinct real solutions.
Therefore, the polynomial function P(x) = -2x^2 + 2x + 4 has two real roots.