Ohio Assessments for Educators (OAE) Mathematics Practice Exam

If the discriminant is negative, what can be inferred about the roots of the quadratic equation?

1. There are two distinct real roots

2. There is one real root

3. There are no real roots

4. All roots are positive

The correct answer is: There are no real roots

When the discriminant of a quadratic equation, which is represented as \(b^2 - 4ac\), is negative, it indicates that the quadratic does not intersect the x-axis at any point. This leads to the conclusion that the equation has no real roots. A negative discriminant means that the solutions to the quadratic equation are complex (or imaginary) numbers. This is because, when attempting to find the roots using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), the square root of a negative number results in an imaginary number. Therefore, the quadratic will have two complex conjugate roots instead of real roots. In this context, while real roots represent values on the number line where the graph of the quadratic touches or crosses the x-axis, a negative discriminant signifies that the graph does not touch or cross the x-axis at all, reinforcing that there are no real roots present.