Completing the Square - Method, Formula, and Examples

A lot of students feel intimidated when they first see quadratic equations. You might be comfortable with basic factoring, but what do you do when a quadratic expression doesn't break down into neat factors? This is where completing the square technique comes to your rescue. It is a reliable method that always works, even when other methods fail. If you know this method, you can turn a confusing equation into one that is easy to read and solve. This helps in creating a firm base in algebra or mental maths when you are preparing for a class test or exam.

What is Completing the Square?

It is a way to rewrite a quadratic expression of the form ax² + bx + c into a form like a(x + d)² + e. This "perfect square" format helps in identifying the roots of the equation or finding the vertex of a parabola easily. 

Consider it like building a puzzle. If you have almost all the pieces needed to make a perfect square but are missing one small corner, you simply calculate what that missing piece is and add it to both sides. This keeps the equation in balance and makes it much easier to read.

What is the Completing the Square Formula?

You need to know this formula to apply this strategy well. The goal is to change the left side of your equation so that it looks like (x + h)².

If you have a quadratic equation in the form x² + bx + c = 0, the logic goes like this:

1. Find the value of b, which is the coefficient of x.

2. To get (b/2), divide this number by 2 .

3. To get (b/2)², square this result.

This value, (b/2)², is the "magic number" you add to the expression to create a perfect square trinomial. In most cases, the resulting identity looks like this:

x² + bx + (b/2)² = (x + b/2)²

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Steps of Completing the Square Method

The easiest method to avoid making mistakes is to follow the series of steps below. Here is the traditional way broken down for the students:

Step 1: Put the terms in order

Ensure your equation looks like ax² + bx + c = 0.

Step 2: Simplify the x² coefficient. 

If a is not 1, divide the entire equation by a. The approach works best when the leading coefficient is 1.

Step 3: Move the constant. 

Shift the constant term (c) to the right side of the equals sign.

Step 4: Find the magic number. 

Find the value of b, which is the coefficient of x, and then divide it by 2 and square it.

Step 5: Add to both sides. 

To keep the equation balanced, add this squared value to both the left and right sides.

Step 6: Factor the left side. 

Now, the left side will now be a perfect square. Change it to (x + b/2)².

Step 7: Solve. 

Take the square root of both sides and find x.

Read More - Ordinal Numbers: Definition, Examples, and Uses in Mathematics

Completing the Square Examples

Let’s look at some examples to see how this works in practice.

Example 1: Solving x² + 4x - 5 = 0

1. Move the constant: x² + 4x = 5

2. Take half of 4 (which is 2) and square it: 2² = 4

3. Add 4 to both sides: x² + 4x + 4 = 5 + 4

4. Simplify: x² + 4x + 4 = 9

5. Factor the left side: (x + 2)² = 9

6. Take the square root: x + 2 = ±3

7. Solve for x: x = 1 or x = -5

Example 2: Solving x² - 6x + 8 = 0

1. Move the constant: x² - 6x = -8

2. Take half of -6 (which is -3) and square it: (-3)² = 9

3. Add 9 to both sides: x² - 6x + 9 = -8 + 9

4. Simplify: (x - 3)² = 1

5. Take the square root: x - 3 = ±1

6. Solve for x: x = 4 or x = 2

Why Is Completing the Square Method Better Than Factoring?

You might be wondering why we don't just factor every equation. Factoring is quick, but it only works with "nice" integers. It lets you to: 

 Solve any quadratic: Even if the roots have decimals or square roots in them.

Find the Quadratic Formula: This is how the famous quadratic formula was formed. 

Find the Vertex: This method helps you identify the highest or lowest point on a curve in geometry.

Read More - Volume of Cubic Foot - Formula, Definition, Examples

Tips for Completing the Square Worksheet

Remember these tips when you are practicing:

• Look for the signs: If your b value is negative, half of it will likewise be less than zero. But, if you square it, the answer will always be positive.

• Keep the scale in balance: Whatever you add to the left side, you must add to the right side immediately.

• Fractional coefficients: Don't be terrified of fractions. If b is 3, then half of it is 3/2, and squaring it gives 9/4. For better accuracy, use fractions instead of decimals. 

Summary  Table for Completing the Square

Here is the quick summary of the method for quick revision:

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