Accuplacer Advanced Algebra and Functions Practice Exam

To determine if a quadratic equation has two real solutions, we can use the discriminant, which is part of the quadratic formula. The discriminant is given by the formula \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation in the standard form \(ax^2 + bx + c = 0\).

For the equation \(x^2 + 5x + 6 = 0\):

1. Identify the coefficients: \(a = 1\), \(b = 5\), and \(c = 6\).

2. Calculate the discriminant:

\[

D = b^2 - 4ac = 5^2 - 4(1)(6) = 25 - 24 = 1.

\]

Since the discriminant \(D = 1\) is greater than zero, this indicates that the quadratic equation has two distinct real solutions.

In contrast, for the other equations:

- The equation \(x^2 + 2x + 3 = 0\) has a negative discriminant (\(D < 0\)), resulting in complex solutions