Final answer:
For quadratic functions, we can determine the number of real and nonreal solutions by calculating the discriminant. If the discriminant is positive, there are two real solutions. If the discriminant is zero, there is one real solution. If the discriminant is negative, there are two nonreal solutions.
Step-by-step explanation:
To determine the number of real and nonreal solutions for each quadratic function, we can look at the discriminant of the quadratic equation. The discriminant, denoted as Δ, is calculated as b^2 - 4ac. If Δ is positive, there are two real solutions. If Δ is equal to zero, there is one real solution. And if Δ is negative, there are two nonreal solutions, also known as complex solutions.
For the given quadratic functions:
A) (x^2 + 4x + 4 = 0)
Here, a = 1, b = 4, and c = 4. Calculating the discriminant: Δ = 4^2 - 4(1)(4) = 0. Since Δ = 0, there is one real solution.
B) (3x^2 - 6x + 9 = 0)
a = 3, b = -6, and c = 9. Δ = (-6)^2 - 4(3)(9) = 0. Again, there is one real solution.
C) (2x^2 + 5x - 3 = 0)
a = 2, b = 5, and c = -3. Δ = 5^2 - 4(2)(-3) = 61. Since Δ > 0, there are two real solutions.
D) (4x^2 + 7x + 2 = 0)
a = 4, b = 7, and c = 2. Δ = 7^2 - 4(4)(2) = 9. Since Δ > 0, there are two real solutions.