Completing the Square | Math Notes | Abdur-Rahmān Bilāl

Completing the square is one of the most elegant techniques in algebra. It transforms a quadratic expression that might look messy and hard to work with into a perfect, neat, squared form. This technique is the very foundation behind the quadratic formula itself.

1. What Does It Mean to Complete the Square?

Let us start with what a perfect square trinomial looks like. When you expand $(x + n)^2$, you get:

$$(x + n)^2 = x^2 + 2nx + n^2$$

$x^2$

The squared term

$2nx$

Middle coefficient is $2n$

$n^2$

Constant term is $n$ squared

This tells us that a perfect square trinomial has a very specific structure. The middle coefficient is always exactly twice the value of $n$, and the constant term is always $n^2$. If you know one piece, you can figure out the other.

The Core Idea

Completing the square is the process of taking a quadratic expression like $x^2 + bx$ and figuring out what constant you need to add to make it a perfect square. The answer is always $\left(\frac{b}{2}\right)^2$, and the resulting factored form is always $\left(x + \frac{b}{2}\right)^2$.

Real-World Analogy: Completing a Puzzle

Imagine you are assembling a jigsaw puzzle. You have most of the pieces, but one piece is missing, and you know exactly what shape it needs to be because the surrounding pieces define the gap. Completing the square works the same way. You have $x^2$ and the $bx$ term, and there is exactly one number that fits perfectly in the remaining slot to make everything form a complete, squared expression. The puzzle piece is always $\left(\frac{b}{2}\right)^2$.

2. The Step-by-Step Process

Given: $x^2 + bx$

Step 1: Take the coefficient of $x$ and divide by 2

The coefficient of $x$ is $b$. Divide it by 2 to get:

$$n = \frac{b}{2}$$

Step 2: Square that result

This gives you the number you add to complete the square:

$$\left(\frac{b}{2}\right)^2$$

Step 3: Write the factored form

The expression $x^2 + bx + \left(\frac{b}{2}\right)^2$ now factors perfectly into:

$$\left(x + \frac{b}{2}\right)^2$$

The Complete Formula

$$x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$$

3. Worked Example

Find $n$: $x^2 + 11x + \frac{121}{4} = (x + n)^2$

We need to find the value of $n$. Looking at the left side, the coefficient of $x$ is $11$, so $b = 11$.

Step 1: Divide $b$ by 2

$$n = \frac{11}{2}$$

Step 2: Verify $n^2$ matches the constant term

$$\left(\frac{11}{2}\right)^2 = \frac{121}{4} \quad \checkmark$$

Step 3: Verify the middle term

$$2n = 2 \cdot \frac{11}{2} = 11 \quad \checkmark$$

Answer:

$$n = \frac{11}{2}$$

Verification Checklist

When completing the square, always verify your answer by checking two things:

Check 1

Does $2n$ equal the coefficient of $x$?

$2 \cdot \frac{11}{2} = 11$ ✓

Check 2

Does $n^2$ equal the constant term?

$\left(\frac{11}{2}\right)^2 = \frac{121}{4}$ ✓

If both checks pass, you have the right answer. If either fails, recheck your arithmetic.

4. Why Completing the Square Matters

You might wonder why this technique is important when we already have the quadratic formula. The answer is that completing the square is actually how the quadratic formula was derived in the first place. Every time you use the quadratic formula, you are essentially completing the square in a generalized way.

Beyond that, completing the square shows up in many areas of higher mathematics:

Vertex Form

Rewrite quadratic functions to reveal the maximum or minimum value.

Calculus

Used for integration techniques involving quadratic expressions.

Engineering

Analyze the behavior of circuits and control systems.

Deriving Formulas

The foundation behind the quadratic formula itself.

Method	When to Use	Strengths
Factoring	When the equation factors into nice whole numbers	Fast and clean, but only works sometimes
Completing the Square	Always works; best when leading coefficient is 1	Gives squared form directly; useful for vertex form
Quadratic Formula	Always works for any quadratic equation	Most general method; the universal fallback

5. Practice Problems

Try completing the square for these expressions. For each one, find the value of $n$ that makes the equation true:

Problem 1:

$$x^2 + 8x + \text{___} = (x + n)^2$$

Solution: $b = 8$, so $n = \frac{8}{2} = 4$

Add $4^2 = 16$ to complete the square.

$$x^2 + 8x + 16 = (x + 4)^2$$

Problem 2:

$$x^2 + 5x + \text{___} = (x + n)^2$$

Solution: $b = 5$, so $n = \frac{5}{2}$

Add $\left(\frac{5}{2}\right)^2 = \frac{25}{4}$ to complete the square.

$$x^2 + 5x + \frac{25}{4} = \left(x + \frac{5}{2}\right)^2$$

Problem 3:

$$x^2 + 20x + \text{___} = (x + n)^2$$

Solution: $b = 20$, so $n = \frac{20}{2} = 10$

Add $10^2 = 100$ to complete the square.

$$x^2 + 20x + 100 = (x + 10)^2$$