Difference of Squares

A difference of squares is a binomial in which one perfect square is subtracted from another, like x² − 9. It factors in one step: a² − b² = (a + b)(a − b). Factoring is the reverse of multiplying, so you can always check an answer by multiplying it back out. In this lesson, “factor” means “factor using integers,” and the pattern needs a minus sign between two perfect squares.

The pattern

a² − b² = (a + b)(a − b)

The two factors use the same two pieces, a and b. They differ only in the sign between them: one plus and one minus. The order of the factors does not matter, because multiplication can be done in either order.

Try it with numbers. For a = 9 and b = 4: a² − b² = 81 − 16 = 65, and (9 + 4)(9 − 4) = 13 × 5 = 65.

The letters a and b can stand for anything you can square: numbers, variables, or terms such as 2x. In x² − 9 we have a = x and b = 3, so x² − 9 = (x + 3)(x − 3).

If you met the product of a sum and a difference in Special Products of Binomials, this is that same fact read from right to left.

Why it works: expand and look at the middle

Multiply (a + b)(a − b) one term at a time:

(a + b)(a − b) = a·a − a·b + b·a − b·b = a² − ab + ab − b² = a² − b²

The two middle terms, −ab and +ab, have the same size and opposite signs, so they cancel. They cancel only because the two factors have opposite signs. In (a + b)(a + b) the middle terms are both +ab and add up to 2ab, so (a + b)² = a² + 2ab + b². That is not a difference of squares.

Why it works: a picture

Start with a big square whose sides are a long. Cut a small square with sides b long out of one corner. (In the picture, a is bigger than b.) The L-shaped piece that is left has area a² − b².

Cut the L into two rectangles: a bottom strip that is a wide and a − b tall, and a piece above it that is a − b wide and b tall. Turn the second piece on its side and slide it to the end of the strip. Together they make one rectangle that is a + b wide and a − b tall. Its area is (a + b)(a − b).

Nothing was added or thrown away, so the two areas are equal: a² − b² = (a + b)(a − b).

The picture needs positive lengths with a > b. The expanding argument above shows the identity is true for all numbers.

Spotting perfect squares

A perfect square is what you get when you square something. With whole-number coefficients, the numbers to know are the squares of 1 through 12: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144.

A term is a perfect square when both of these are true:

• its coefficient is a perfect square (a lone variable power such as x⁶ has coefficient 1, which counts), and
• every variable in it has an even exponent.

To find the thing that was squared, take the square root of the coefficient and halve each exponent. Halving works because squaring doubles an exponent: (x³)² = x³ · x³ = x⁶.

• 1 = 1²
• 49 = 7²
• x⁶ = (x³)²
• 9y² = (3y)²
• 16x⁴ = (4x²)²

Not perfect squares: 20x² (20 is not a perfect square), x⁵ (the exponent is odd) and 7.

Put it together with 4x² − 25: 4x² = (2x)² and 25 = 5², so a = 2x and b = 5. Then 4x² − 25 = (2x + 5)(2x − 5). Check by expanding: 4x² − 10x + 10x − 25 = 4x² − 25.

The steps

1. Take out any GCF (greatest common factor) of all the terms first.
2. Check the shape. You need exactly two terms, and each must be a perfect square, with one subtracted from the other. It does not matter which square is written first: 100 − 9t² qualifies.
3. Find a and b, the things that were squared.
4. Write (a + b)(a − b).
5. Check by expanding. The middle terms should cancel and you should land back on the start.
6. Look again. If any factor can still be factored, keep going until the expression is completely factored.

Take out the GCF first

Sometimes the squares are hiding behind a common factor. In 20x² − 5, neither 20 nor 5 is a perfect square, so it does not look like a difference of squares. But 5 divides both terms: 20x² − 5 = 5(4x² − 1). Now 4x² = (2x)² and 1 = 1², so 20x² − 5 = 5(2x + 1)(2x − 1). The GCF stays out front of the finished product.

Looking for a GCF first is a strategy, not a guarantee. After you take it out, the terms still have to be perfect squares. For 3x² − 20 the GCF is 1 and neither term is a perfect square, so this pattern does not apply.

Completely factored: keep going

An expression is completely factored when you have taken out any GCF and every other factor is prime, meaning it cannot be factored any further using integers.

Look at x⁴ − 16. Since x⁴ = (x²)² and 16 = 4², it is a difference of squares with a = x² and b = 4:

x⁴ − 16 = (x² + 4)(x² − 4)

Seeing x⁴ − 16 as (x²)² − 4² means reading the structure of the expression: the same pattern can sit inside bigger pieces.

Now inspect each factor. The factor x² − 4 = x² − 2² is another difference of squares, so it factors again: (x + 2)(x − 2). The factor x² + 4 is a sum of squares and cannot be factored (see the next section). So

x⁴ − 16 = (x² + 4)(x + 2)(x − 2)

Stopping at (x² + 4)(x² − 4) is a correct factorization, but it is not complete, because x² − 4 can still be factored.

Why a sum like x² + 9 does not factor

Compare x² − 9 and x² + 9. The first is (x + 3)(x − 3). The second is a sum of squares, and the pattern needs a minus sign.

Try to force it. (x + 3)(x − 3) = x² − 9 has the wrong sign. (x + 3)(x + 3) = x² + 6x + 9 and (x − 3)(x − 3) = x² − 6x + 9 have a middle term. To make the middle terms cancel, the signs must be opposite, and opposite signs make the last term negative. So no choice of signs gives x² + 9.

That only tried factors that look like (x ± 3). Here is an argument for all numbers. Any way of writing x² + 9 as a product of two binomials with real coefficients can be put in the form (x + p)(x + q), because the x² term has coefficient 1. Put x = −p. The right side becomes 0. The left side becomes p² + 9, which is at least 9, never 0. So no real numbers p and q work. The same idea works for any sum like 4x² + 25.

So x² + 9 is prime: it cannot be factored using integers, and it cannot be factored with any real-number coefficients either. (A later course uses imaginary numbers, with i² = −1, to write x² + 9 = (x + 3i)(x − 3i).)

A sum of squares can still have a GCF: 4x² + 36 = 4(x² + 9). The GCF 4 comes out, and the x² + 9 stays as it is.

A caution about how far this goes: this section is about two-term sums of degree 2, like x² + 9 and 4x² + 25. A few sums with higher powers, such as x⁴ + 4 = (x² + 2x + 2)(x² − 2x + 2), can be factored by methods beyond this lesson.

Mental math

The pattern also multiplies numbers quickly. If two numbers are the same distance from a number whose square you know, one is a sum and the other is a difference: (n + d)(n − d) = n² − d².

For 47 × 53: both numbers are 3 away from 50, so 47 × 53 = (50 − 3)(50 + 3) = 50² − 3² = 2500 − 9 = 2491.

It works only when the numbers sit the same distance on either side of the middle number. A pair like 47 × 52 is not the same distance from 50, so the trick does not apply with 50 as the middle.