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What is a Perfect Square in Math?

What is a perfect square? It is a number that, when multiplied by itself, equals another number.
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What is a Perfect Square in Math?
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Picture a perfect square. It has four perfect right angles, and it's perfectly symmetrical. Everything is in order, and nothing is out of place.

It makes you remember geometry class, right? All that talks about squares and angles, shapes, and other numerical values.

But what is a perfect square, exactly? What does this concept mean, and why are they important in math? 

In this article, we will discuss everything about perfect squares, including what they are and how to identify them.

What is a Perfect Square?

A perfect square is a specific type of number called a square number. Most of the time, a square number is the sum of two integers that are both the same size.

So, 1, 4, 9, 16, and 25 are all square numbers because they can be expressed as 1 x 1, 2 x 2, 3 x 3, 4 x 4, and 5 x 5.

Think of physical objects making up a perfect square, like the tiles on a bathroom floor. There are always an equal number of tiles on each side, and the length and width are always the same when arranged in this sequence.

Or three rows of apples in three places. You tend to get a perfect square when you arrange objects like this, and that's what you're picturing when you think of a square number.

4 is a square number because 2 x 2 = 4

9 is a square number because 3 x 3 = 9

16 is a square number because 4 x 4 = 16

25 is a square number because 5 x 5 = 25

36 is a square number because 6 x 6 = 36

And so on. As you can see, the square numbers follow a pattern.

So, 1, 4, 9, 16, and 25 are all square numbers because they can be expressed as 1 x 1, 2 x 2, 3 x 3, 4 x 4, and 5 x 5.

Note that a perfect square must be a positive integer—You can't have a negative perfect square or a fractional perfect square. In other words, numbers like 5.5 or 4/5 cannot be expressed as the product of two equal integers, so they're not perfect squares.

Perfect Square Formula—How to Find a Perfect Square

Like many other mathematical concepts, perfect squares can be expressed using a formula.

If x is any whole number, the square of x can be written like this:

x² = x × x

For instance, using this formula, we can calculate that 12² = 12 × 12 = 144.

Similarly, we can use the formula to calculate that 30² = 30 × 30 = 900.

In the above examples, the perfect squares are the products (the answers), and the numbers we started with (12 and 30) are called the square roots of those perfect squares.

The square root of a number is just the number that, when multiplied by itself, gives you the original number. So the square root of 144 is 12 because when you multiply 12 by 12, the total number you get is 144.

And the square root of 900 is 30 because 30 × 30 = 900

Another way to find perfect squares is by finding the square root of the given number.

Perfect square roots

You always have a corresponding perfect square root and a perfect square. Perfect square roots are the numbers that, when squared, give you the perfect squares.

So, as we've seen, the square root of 144 is 12. The perfect square root of 900 is 30.

And the square root of 4 is 2 because 2 × 2 = 4.

The square roots of perfect squares are always whole numbers, and that's because you can't have a fractional or decimal number when you're multiplying two numbers together.

You can get the perfect square root by dividing the perfect square with

Table of Perfect Squares

When we talk about perfect squares, we usually mean the square of a whole number. So 1, 4, 9, 16, 25, and so on are all perfect squares.

But there are other perfect squares besides these. In fact, there are an infinite number of them! The following table shows a list of some of the most well-known ones:

Natural Whole Numbers (Square roots) Perfect Square Numbers
1² = 1 = 1
2² = 4 = 4
3² = 9 = 9
4² = 16 = 16
5² = 25 = 25
6² = 36 = 36
7² = 49 = 49
8² = 64 = 64
9² = 81 = 81
10² = 100 = 100
11² = 121 = 121
12² = 144 = 144
13² = 169 = 169
14² = 196 = 196
15² = 225 = 225
16² = 256 = 256
17² = 289 = 289
18² = 324 = 324
19² = 361 = 361
20² = 400 = 400

As you can see, the square of any whole number is a perfect square.

What is the Perfect Square Trinomial?

In mathematics, a trinomial is an expression that has three terms. So, for example, the expression x² + 5x + 6 is a trinomial because it has three terms: x², 5x, and 6.

A perfect square trinomial comes from the square of two binomials (expressions with two terms). It's the square of two binomials.

Let's look at other examples

Binomial—(x + 2)

We square this binomial to get a perfect square trinomial:

(x + 2)²

= (x + 2) x (x + 2)

= x² + 4x + 4

Do you see how we've taken the square of the binomial (x + 2) to get a trinomial?

The first term in the trinomial is x², the second term is 4x, and the third term is 4. Each of these terms comes from squaring the terms in the original binomial.

The term x² comes from squaring x, the term 4x comes from squaring 2 and then multiplying by x, and the term 4 comes from squaring 2.

Let's look at another example:

Binomial—(x - 3)

We square this binomial to get a perfect square trinomial:

(x - 3)²

= (x - 3) x (x - 3)

= x² - 6x + 9

As before, we've taken the square of the binomial (x - 3) to get a trinomial.

You can use this method to square any binomial and get a perfect square trinomial. All you need to do is square each term in the binomial and then add them together.

How to Identify Perfect Squares

Identifying a perfect square is often just a matter of recognizing a pattern. For example, the perfect squares we listed earlier all have a few things in common:

  • They usually have 0, 1, 4, 5, 6, or 9 in their units digit.
  • The last two digits are either the same number (44, 55, 66, 99) or add up to 9 (00, 11, 24, 36, 49).
  • The square of an odd number always ends with an odd number. For example, 9² = 81 and 3² = 9.
  • The square of an even number always ends in an even  number like 4, or 6. For example, 4² = 16 and 12² = 144.
  • A number that ends with 5 will have 5 at the units place digit when squared. For example, 25² = 625 and 5² = 25.

Keep these patterns in mind, and you'll be able to quickly identify perfect squares when you see them.

Of course, not all perfect squares will fit this pattern. But many of them will. So if you're ever unsure whether a number is a perfect square, these are some things you can check for.

In conclusion, a perfect square is a whole number that can be expressed as the square of another whole number. It also has a corresponding perfect square root, which, when squared, gives you the perfect square.

All perfect squares are whole numbers, and any whole number can be squared to get a perfect square. Also, remember that the square of a binomial is known as the perfect square trinomial.

Perfect squares often conform to specific patterns, but not always. The best way to determine whether a number is a perfect square is to find the square root of the number and see if it's a whole number.

Get in touch with us at Learner to learn more about perfect squares and other mathematical concepts. Start your online lesson today. Our tutors will schedule your lessons based on your convenience.

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About the author:

Mike developed his passion for education as a math instructor at Penn State University. He expanded his educational experience launching and running an Executive Education business - training over 100,000 students per year. As the CEO of Learner, Mike focuses on accelerating learning and unleashing the potential of students. 

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