Lesson 1.20

The Quadratic Formula

The "Universal Solving Machine." If you can't factor it, and you don't want to complete the square, this formula will ALWAYS find the answer.

Introduction

By completing the square on the generic equation , mathematicians derived a single formula that works for everything.

Past Knowledge

You've solved by factoring and square roots.

Today's Goal

Memorize and apply the formula to solve equations involving .

Future Success

This formula is programmed into calculators and computers to solve projectile motion problems instantly.

Key Concepts

1. The Formula

Where (The Discriminant)

Calculate first. It makes the formula much less scary!

2. Standard Form Requirement

The equation MUST be equal to zero before you start.

If it equals 5, you must subtract 5 first!

Worked Examples

Example 1: Two Real Solutions

Basic

Solve .

1

Find the Discriminant

2

Plug D into Formula

Example 2: Complex Solutions

Intermediate

Solve .

1

Find the Discriminant

2

Plug D into Formula

Example 3:

Advanced

Solve .

1

Find the Discriminant

Watch the signs! .

2

Plug D into Formula

Common Pitfalls

The Radical Line

The entire numerator () must be divided by . Do not just divide the radical part.

Negative

If , then . Also is always positive ().

Real-Life Applications

This is the "hammer" of algebra. In ballistics (calculating when a rocket hits the ground), the numbers are rarely clean integers. We use the formula to solve to find impact times to within milliseconds.

Practice Quiz

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