Lesson 1.19

The Discriminant

Sometimes you don't need the exact solution; you just need to know how many solutions exist. The Discriminant tells the future without doing the hard work.

Introduction

Look inside the Quadratic Formula at the part under the square root: . This number determines everything about the roots.

Past Knowledge

is Real. . is Imaginary.

Today's Goal

Calculate and classify the roots.

Future Success

This visualizes whether a parabola intersects the x-axis twice, once, or never.

Key Concepts

1. The Three Cases

Positive ()

Two distinct real solutions. (Intersects x-axis twice).

Zero ()

One real solution. (Vertex touches x-axis).

Negative ()

Two complex conjugate solutions. (Does not touch x-axis).

2. Why It Works

Looking at the formula:

  • If , we add/subtract 5 (Two answers).
  • If , we add/subtract 0 (One Answer).
  • If , we get (Imaginary).

Worked Examples

Example 1: Positive Discriminant

Basic

Classify roots for .

1

Calculate D

2

Interpret

. Positive.

Result: 2 Real Solutions.

Example 2: Zero Discriminant

Intermediate

Classify roots for .

1

Calculate D

2

Interpret

Result: 1 Real Solution.

(This means it was a perfect square trinomial!)

Example 3: Negative Discriminant

Advanced

Classify roots for .

1

Calculate D

2

Interpret

. Negative.

Result: 2 Complex Solutions.

Common Pitfalls

Sign Errors

In , watch out when or are negative. becomes .

Don't Square Root It!

The Discriminant is just the number inside. Do not take the square root of to classify.

Real-Life Applications

In physics, if we calculate the trajectory of a rocket, the Discriminant tells us if it will ever hit the ground () or if it will explode in mid-air and never reach zero height ().

Practice Quiz

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