Properties of Quadratics Determining the “Nature of the Roots” We know that the roots are found where the quadratic equation intersects the x-axis. We now need to determine the nature of the roots. In other words, describe what kind of roots we have. To do that, we need to use part of the quadratic formula: We are going to use the part of the quadratic formula under the radical. This is known as the DISCRIMINANT.

Using the discriminant: There are three things we need to check for: 1.Real vs. Imaginary 2.Equal vs. Unequal 3.Rational vs. Irrational Lets check the four cases: Case 1: Discriminant is negative: This means that the roots are imaginary because we have a negative under the radical. Case 2: Discriminant is zero: This means that the roots are real, equal and rational. EX: Suppose the formula gave us the following: So we see that both roots are ½, so they are equal, and ½ is a real, rational number!

Case 3: Discriminant is positive and a perfect square: This means that the roots are real, unequal and rational. EX: Suppose the formula gave us the following: So we see that both roots real, rational numbers that are NOT equal Case 4: Discriminant is positive and not a perfect square: This means that the roots are real, unequal and irrational. EX: Suppose the formula gave us the following: So we see that both roots real, irrational numbers that are NOT equal

Summary: 1. Discriminant is negative so roots are imaginary. ONLY TIME THAT THE ROOTS ARE IMAGINARY! 2. Discriminant is zero so roots are real, rational and equal. ONLY TIME THAT THE ROOTS ARE EQUAL! 3. Discriminant is a positive perfect square so roots are real, rational and unequal. 4. Discriminant is a positive non-perfect square so roots are real, irrational and unequal. ONLY TIME THE ROOTS ARE IRRATIONAL!

Page 1 #1: The discriminant is a positive, non-perfect square. Therefore the roots are: 1. Real 2. Irrational 3. Unequal #2: The discriminant is a positive, perfect square. Therefore the roots are: 1. Real 2. Rational 3. Unequal

Page 1 #5: The discriminant is zero. Therefore the roots are: 1. Real 2. Rational 3. Equal #7: The discriminant is negative. Therefore the roots are: 1. Imaginary

Page 1 #8: The discriminant is a positive, perfect square. Therefore the roots are: 1. Real 2. Rational 3. Unequal #22: The discriminant is a positive, non-perfect square. Therefore the roots are: 1. Real 2. Irrational 3. Unequal

Homework Page 1 #3,6,9,11,13,15,17,19,21,24,27,30,33