Completing the Square

Lesson Plan - Get It!

Audio:

Rocket Math in Action

Imagine your class is launching model rockets during a STEM competition. One rocket’s height is modeled by this equation:

h(x) = -16x² + 96x - 44

In this equation:

• h(x) represents the rocket’s height
• x represents time in seconds

Your challenge:

How high will the rocket go before it starts falling back down?

The answer is hidden inside the quadratic expression. Completing the square helps uncover it.

Why Vertex Form Matters

Quadratic equations can be written in several forms, but one form is especially useful when finding maximums and minimums.

That form is called vertex form:

The vertex of the parabola is the point:

(h, k)

This point tells you:

• the highest point of the graph if the parabola opens downward
• the lowest point of the graph if the parabola opens upward

Quick reminder:

• If a is positive, the parabola opens upward.
• If a is negative, the parabola opens downward.

That means the rocket equation above has a maximum height because the leading coefficient is negative.

What Does “Completing the Square” Actually Mean?

The phrase sounds strange at first, but it comes from geometry.

Imagine building a perfect square using algebra tiles or puzzle pieces. Completing the square means rearranging parts of a quadratic expression to form a perfect square.

That perfect square can then be rewritten in vertex form.

Here is a quadratic expression in standard form:

x² + 8x - 2

Your goal is to rewrite it in vertex form.

Step 1: Focus on the x-Term

Take the coefficient of the x-term.

In this case, the coefficient is 8.

Divide it by 2:

Step 2: Square the Result

Now square the number:

4² = 16

That 16 helps create a perfect square trinomial.

Step 3: Build the Perfect Square

Rewrite the quadratic using the number you found:

(x + 4)²

Why does this work?

Because expanding the expression gives:

(x + 4)² = x² + 8x + 16

The first two terms match perfectly. Only the constant changed.

The original expression ended with - 2, not +16.

So adjust the expression:

x² + 8x - 2 = (x + 4)² - 18

That is the completed square form.

The vertex is:

(-4, -18)

A Faster Way to Think About It

Many students memorize long procedures and get lost halfway through. A faster strategy is to think about matching patterns.

Example:

x² - 10x + 5

Half of -10 is -5.

So start with:

(x - 5)²

Expanding gives:

(x - 5)² = x² - 10x +25

But the original expression has +5, not +25.

Adjust by subtracting 20:

x² - 10x + 5 = (x - 5) ² - 20

This shortcut works because the first two terms always match when you:

1. Take half of the x-coefficient.

2. Place it inside the parentheses.

3. Square it.

What Happens When There Is a Leading Coefficient?

Things get slightly trickier when a number appears in front of x².

Example:

Many students panic at this point. No need. The process is almost the same.

First, factor out the leading coefficient from the x-terms:

Now complete the square inside the parentheses.

Half of -5 is:

Square it:

Add and subtract the value inside the parentheses:

Rewrite the perfect square:

The key idea: 

Remove the leading coefficient first. Then complete the square normally.

Using Vertex Form to Solve Real Problems

Return to the rocket equation:

h(x) = -16x² + 96x - 44

After completing the square, the equation becomes:

h(x) = - 16(x - 3)² + 100

The vertex is:

(3, 100)

That means:

• The rocket reaches its maximum height after 3 seconds.
• The maximum height is 100 feet.

This is why completing the square matters. It transforms a messy quadratic into a form that instantly reveals important information.

Common Mistakes to Avoid

1. Forgetting to divide the x-coefficient by 2 before squaring

2. Forgetting to adjust the constant afterward

3. Mixing up the vertex signs

4. Forgetting to factor out the leading coefficient first

5. Expanding incorrectly when checking your answer

Helpful Tip:

Always expand your final answer to verify that it matches the original quadratic.

For example:

-16(x - 3)² + 100 = - 16x² + 96x - 44

You are now ready to practice completing the square yourself.

In the Got It? section, you will work through guided problems, check your thinking step by step, and build confidence before tackling real-world applications.