Perfect squares

Perfect squares are the numbers you get when you multiply a whole number by itself: 1, 4, 9, 16, 25, 36 and so on. For example, 49 is a perfect square because 7 × 7 = 49. The square root of a perfect square is always a whole number, and that is the quickest test: √49 = 7, but √50 isn't a whole number, so 50 is not a perfect square.

List of perfect squares from 1 to 900

There are 10 perfect squares from 1 to 100 and 31 from 0 to 900, if you count 0 = 0². Learning the first 15 by heart pays off: AQA's GCSE Maths specification includes square numbers up to 15 × 15, and US grade 8 Common Core asks students to evaluate square roots of small perfect squares. Perfect squares are also called square numbers.

Why they are called squares

Arrange 16 dots in rows and you can make a 4 by 4 square. Try 15 dots and you can't: a square needs the same number in every row and column. So the perfect squares are exactly the numbers of dots, tiles or counters that make a square. The area of a square with whole-number sides is always a perfect square.

How to tell if a number is a perfect square

1. Check the last digit

Perfect squares only end in 0, 1, 4, 5, 6 or 9. A number ending in 2, 3, 7 or 8 is never a perfect square, so 1,447 and 5,328 are out straight away. A number ending in 1, 4, 5, 6, 9 or 0 might be one, so go to the next test.

2. Check the zeros

A perfect square ending in zeros has an even number of them: 100, 3,600, 250,000. So 2,000, with three zeros, is not a perfect square.

3. Use prime factors

A number is a perfect square exactly when every prime in its prime factorisation has an even power. 1,764 = 2² × 3² × 7², all even, so it is a perfect square: 2 × 3 × 7 = 42, and 42² = 1,764. But 72 = 2³ × 3² has an odd power of 2, so it is not. The prime factorisation calculator does the splitting for you.

4. Estimate the square root

Find the perfect squares either side. For 5,776: 70² = 4,900 and 80² = 6,400, so a root would be between 70 and 80. It ends in 6, so the root ends in 4 or 6. Try 76: 76² = 5,776. It is a perfect square.

Patterns in perfect squares

• The gaps are the odd numbers. 1, 4, 9, 16, 25 go up by 3, 5, 7, 9. So the first n odd numbers add up to n²: 1 + 3 + 5 + 7 + 9 = 25.
• The next square: (n + 1)² = n² + 2n + 1. If you know 30² = 900, then 31² = 900 + 60 + 1 = 961.
• Squares ending in 5: multiply the front digit by the next number up, then write 25. For 65², 6 × 7 = 42, so 65² = 4,225.
• Even and odd: even numbers have even squares, and odd numbers have odd squares.

Perfect square trinomials

In algebra, a perfect square trinomial is a quadratic that is a bracket squared. There are two patterns:

To spot one, check three things: the first term is a square, the last term is a square, and the middle term is twice the product of their square roots.

• x² + 6x + 9: x² and 9 = 3² are squares, and 2 × x × 3 = 6x. So it is (x + 3)².
• x² − 10x + 25: 25 = 5², and 2 × x × 5 = 10x, with a minus sign. So it is (x − 5)².
• 4x² + 12x + 9: 4x² = (2x)² and 9 = 3², and 2 × 2x × 3 = 12x. So it is (2x + 3)².
• x² + 8x + 12: 12 isn't a perfect square, so this is not a perfect square trinomial. It factors as (x + 2)(x + 6).

Try factoring one yourself, then press Mark to check it in the course. Perfect square trinomials are also the idea behind completing the square.

Where perfect squares are useful

• Simplifying square roots: split out the biggest perfect square factor. √72 = √36 × √2 = 6√2.
• Estimating square roots: √60 is between √49 = 7 and √64 = 8.
• Pythagoras' theorem: 9 + 16 = 25 gives the 3, 4, 5 right-angled triangle.
• Difference of two squares: a² − b² = (a + b)(a − b). It makes mental maths quick: 49 × 51 = (50 − 1)(50 + 1) = 2,500 − 1 = 2,499.

Common mistakes

• Doubling instead of squaring. 6² is 36, not 12.
• Trusting the last digit alone. Ending in 1, 4, 5, 6, 9 or 0 means a number might be a square, not that it is. 21 ends in 1 but isn't one.
• Forgetting 0 and 1. Both are perfect squares: 0 × 0 and 1 × 1.
• Missing the middle term check for trinomials. x² + 5x + 9 has square ends but isn't (x + 3)², because the middle would need to be 6x.

To practise number skills among other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 paper. For square roots in more depth, see the square root guide.

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