Lesson 1.17

Completing the Square (The Pattern)

Before we solve equations, we need to master a special pattern. We can force any quadratic to become a perfect square trinomial.

Introduction

A "perfect square trinomial" works like this: . But what if you only have ? Today, we learn how to find that missing number (the 9) to complete the pattern.

Past Knowledge

You know .

Today's Goal

Find the value that makes a perfect square.

Future Success

This is the first step to deriving the Quadratic Formula!

Key Concepts

1. The Magic Number Formula

To complete the square for , add:

"Half of , then squared."

2. Factoring the Result

Once you add the magic number, it ALWAYS factors into:

The number inside the parenthesis is just half of .

Worked Examples

Example 1: Even Middle Term

Basic

Complete the square for .

1

Find Magic Number

. Cut it in half (). Square it ().

2

Write Perfect Square

Example 2: Negative Term

Intermediate

Complete the square for .

1

Find Magic Number

. Half is . Square is .

2

Factor

Note: The sign matches the middle term.

Example 3: Odd Middle Term

Advanced

Complete the square for .

1

Find Magic Number

. Half is . Square it.

2

Factor

Don't panic! It factors to .

Common Pitfalls

Forgetting to Square

For , students often add (the half) instead of (the square).

Sign Confusion

The term added is ALWAYS positive. .

Real-Life Applications

While "Completing the Square" sounds abstract, it's actually the mathematical engine behind converting Standard Form () into Vertex Form (). This lets us instantly find the maximum height of a projectile or the optimal price for a product.

Practice Quiz

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