Do I have to memorize the special product formulas, or can I just use FOIL every time?

FOIL will always give you the right answer, so nothing is lost mathematically. But the patterns matter for recognition, not just speed. In the next lessons you factor expressions like x^2 - 121 and 4x^2 + 20x + 25, and to do that you have to spot the pattern in reverse. Students who only ever distribute tend to miss those structures. Learn the patterns, and use FOIL as a check when you are unsure.

Is (a-b)^2 the same as (b-a)^2?

Yes. Expanding (b-a)^2 gives b^2 - 2ab + a^2, which is the same three terms as a^2 - 2ab + b^2 in a different order. This makes sense because b - a is the opposite of a - b, and squaring any number and its opposite gives the same result. Note that this does not hold without the square: a - b and b - a are not equal unless a = b.

Why doesn't x^2 + 25 factor?

The difference of squares pattern requires subtraction, because that is what makes the cross terms cancel. If you try (x+5)(x+5) you get x^2 + 10x + 25, and (x+5)(x-5) gives x^2 - 25. No pair of real binomials multiplies to x^2 + 25. A sum of two squares is prime over the real numbers, and recognizing that quickly is useful in the factoring lessons ahead.

When can I use the mental math shortcut for multiplying two numbers?

Use the difference of squares whenever the two numbers are the same distance above and below an easy anchor, usually a multiple of ten or a hundred: 58 × 62 works because both are 2 from 60. Use the squaring pattern whenever you need the square of a number close to a round one, like 103^2 = 10000 + 600 + 9. If the two factors are not symmetric about a round number, the difference of squares will not apply and you should multiply normally.

ALG1-7.2

Special Products

Master the square of a sum, square of a difference, and difference of squares — plus how to use these patterns to multiply numbers like 47 times 53 in your head.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Special Products, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Special Products free

What this lesson covers

Multiplying two binomials the long way works every time, but three particular products show up so often that mathematicians gave them names and memorized their shapes. Once you can see

(a+b)^2

(a-b)^2

, and

(a+b)(a-b)

coming, you can write the answer in one step instead of four.

These are called special products. They matter for two reasons. First, speed: you will meet them constantly in the factoring lessons that follow, in quadratics, and later in completing the square. Second, structure: recognizing that

x^2 - 49

is really

x^2 - 7^2

is the same skill as recognizing that

47 \times 53

is really

(50-3)(50+3)

. In this lesson you will derive each pattern, learn exactly which sign goes where, see the single most common mistake students make with squaring, and practice turning ugly arithmetic into mental math.

Squaring a Binomial: Why There Are Three Terms

Start with

(a+b)^2

. The exponent means multiply the binomial by itself, so

(a+b)^2 = (a+b)(a+b)

. Distributing gives

a^2 + ab + ba + b^2

, and since

ab

and

ba

are like terms, they combine:

(a+b)^2 = a^2 + 2ab + b^2

The same work with a minus sign gives

(a-b)(a-b) = a^2 - ab - ba + b^2

, so

(a-b)^2 = a^2 - 2ab + b^2

Notice what changes and what does not. The first and last terms are always the squares of the two pieces, and both are always positive (a negative number squared is positive). Only the middle term carries the sign of the original binomial.

Read the pattern in words: square the first term, double the product of the two terms, square the last term. Saying it out loud while you write keeps you from dropping the middle term.

Example:

(x+6)^2 = x^2 + 2(x)(6) + 6^2 = x^2 + 12x + 36

.

Example with a coefficient:

(4m - 3)^2 = (4m)^2 - 2(4m)(3) + 3^2 = 16m^2 - 24m + 9

. The parentheses around

4m

matter — you square the entire term, coefficient included, so you get

16m^2

, not

4m^2

.

Example with two variables:

(2p + 5q)^2 = 4p^2 + 20pq + 25q^2

.

The result of squaring a binomial is called a perfect square trinomial, and its three terms always fit the same fingerprint: two perfect squares on the ends, and a middle term equal to twice the product of their square roots.

The Difference of Squares

Now multiply a sum by the matching difference:

(a+b)(a-b) = a^2 - ab + ba - b^2

. The two middle terms are opposites, so they cancel and leave only two terms:

(a+b)(a-b) = a^2 - b^2

This is the difference of squares. It is the only special product that loses its middle term, and it happens precisely because the inner and outer products are negatives of each other.

For the pattern to apply, the two binomials must have identical terms with only the sign between them different.

(x+9)(x-9)

qualifies.

(x+9)(x-8)

does not. The order of the factors does not matter:

(x-9)(x+9)

gives

x^2 - 81

just the same.

Examples:

Product	Result
$(x+9)(x-9)$	$x^2 - 81$
$(3t-7)(3t+7)$	$9t^2 - 49$
$(5 + 2y)(5 - 2y)$	$25 - 4y^2$
$(ab - 6)(ab + 6)$	$a^2b^2 - 36$

Always subtract the square of the second term from the square of the first term as they appear in the sum-first arrangement. In the third row, writing

4y^2 - 25

would be wrong — the

2y

is the subtracted piece, so its square is what gets subtracted.

One caution: a sum of squares such as

x^2 + 81

does not factor over the real numbers. Only the difference splits. Keeping that straight now will save trouble in the Factoring Special Forms lesson, where you run these same patterns backwards.

The Mistake Almost Everyone Makes Once

The single most common error in this unit is writing

(a+b)^2 = a^2 + b^2

. Squaring is not distributive over addition, and you can disprove the shortcut with numbers in five seconds:

(3+4)^2 = 7^2 = 49

, while

3^2 + 4^2 = 9 + 16 = 25

. The missing

24

is exactly the

2ab

term,

2(3)(4)

.

Why is it so tempting? Because squaring is distributive over multiplication:

(3x)^2 = 9x^2

really does work. The rule splits across products, never across sums or differences. If you are unsure in the moment, rewrite

(a+b)^2

(a+b)(a+b)

and distribute — the long way always tells the truth.

A few more traps worth naming:

Wrong	Right	What went missing
$(x-5)^2 = x^2 - 25$	$x^2 - 10x + 25$	The $-2ab$ middle term, and $(-5)^2$ is $+25$
$(2x+3)^2 = 2x^2 + 12x + 9$	$4x^2 + 12x + 9$	The coefficient was not squared
$(x-4)(x+4) = x^2 + 16$	$x^2 - 16$	Sign: it is $a^2 - b^2$
$-(x+3)^2 = (-x-3)^2$	$-(x^2+6x+9)$	The negative is not inside the square

That last row deserves attention. In

-(x+3)^2

, the exponent applies only to

(x+3)

; expand first, then negate every term to get

-x^2 - 6x - 9

.

When you finish an expansion, check the middle term against the ends: is it twice the product of the square roots of the first and last terms? If not, something slipped.

Using the Patterns for Mental Arithmetic

The patterns are statements about numbers, not just letters, so they turn awkward multiplications into easy ones. The trick is to rewrite each factor in terms of a nearby round number.

For a product of two numbers equally spaced around a round number, use the difference of squares. To find

47 \times 53

, notice both are

3

away from

50

47 \times 53 = (50-3)(50+3) = 50^2 - 3^2 = 2500 - 9 = 2491

Similarly

98 \times 102 = 100^2 - 2^2 = 10000 - 4 = 9996

, and

19 \times 21 = 400 - 1 = 399

.

For a square, use

(a \pm b)^2

. To find

52^2

, write it as

(50+2)^2 = 2500 + 2(50)(2) + 4 = 2500 + 200 + 4 = 2704

. To find

97^2

, go down from

100

(100-3)^2 = 10000 - 600 + 9 = 9409

Problem	Rewrite	Compute
$41 \times 39$	$(40+1)(40-1)$	$1600 - 1 = 1599$
$61^2$	$(60+1)^2$	$3600 + 120 + 1 = 3721$
$95^2$	$(100-5)^2$	$10000 - 1000 + 25 = 9025$
$204 \times 196$	$(200+4)(200-4)$	$40000 - 16 = 39984$

Choose the round anchor first — usually a multiple of

10

— then read off how far each factor sits from it. If the two distances are not equal, the difference of squares does not apply, though the squaring pattern still can. This is the same algebraic structure you have been expanding all lesson, just with digits standing in for

a

and

b

Key terms

Special product.: A product of binomials whose expansion follows a memorized pattern, specifically $(a+b)^2$ , $(a-b)^2$ , and $(a+b)(a-b)$ .
Perfect square trinomial.: A trinomial that results from squaring a binomial, of the form $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$ .
Difference of squares.: An expression of the form $a^2 - b^2$ , which equals the product $(a+b)(a-b)$ .
Binomial.: A polynomial with exactly two terms, such as $3x - 7$ .
Conjugates.: A pair of binomials that differ only in the sign between their terms, such as $2x + 5$ and $2x - 5$ ; their product is a difference of squares.
Middle term.: In a perfect square trinomial, the term $2ab$ or $-2ab$ formed by the two cross products when the binomial is distributed.
Expand.: To rewrite a product as an equivalent sum of terms by multiplying out and combining like terms.

Worked example

Expand

(5x - 4y)^2

and

(5x - 4y)(5x + 4y)

. Then use one of these patterns to compute

96 \times 104

mentally.

Start with the square. Identify the pieces:

a = 5x

and

b = 4y

, and the sign inside is a minus, so use

(a-b)^2 = a^2 - 2ab + b^2

.

Square the first term:

(5x)^2 = 25x^2

. Square the coefficient and the variable together — this is where the most errors happen.

Double the product of the terms:

2(5x)(4y) = 40xy

. Because the binomial has a minus sign, this term is subtracted.

Square the last term:

(4y)^2 = 16y^2

, and it is positive.

So

(5x - 4y)^2 = 25x^2 - 40xy + 16y^2

. Check the fingerprint: the ends are

25x^2

and

16y^2

, whose square roots are

5x

and

4y

; twice their product is

40xy

, matching the middle term.

Now the second product. The two binomials have identical terms and opposite signs, so they are conjugates and the pattern

(a+b)(a-b) = a^2 - b^2

applies. The cross products

+20xy

and

-20xy

cancel, leaving

(5x - 4y)(5x + 4y) = 25x^2 - 16y^2.

For the arithmetic, look for a round anchor. Both

96

and

104

sit exactly

4

away from

100

, so they are conjugates with

a = 100

and

b = 4

96 \times 104 = (100-4)(100+4) = 100^2 - 4^2 = 10000 - 16 = 9984.

Practice questions

Which expression is equivalent to

(3x - 8)^2

$9x^2 - 64$
$9x^2 + 64$
$9x^2 - 48x + 64$
$3x^2 - 48x + 64$

Answer: $9x^2 - 48x + 64$

Use

(a-b)^2 = a^2 - 2ab + b^2

with

a = 3x

and

b = 8

. Squaring the first term gives

(3x)^2 = 9x^2

— the coefficient gets squared too, which rules out the last choice. The middle term is

-2(3x)(8) = -48x

; forgetting it entirely produces the first choice, the classic

(a-b)^2 = a^2 - b^2

error. The final term is

(-8)^2 = +64

. So the expansion is

9x^2 - 48x + 64

Explain why

(a+b)(a-b)

has only two terms in its expanded form, while

(a+b)(a+b)

has three. Then use the shorter pattern to compute

73 \times 67

without a calculator.

Answer: The inner and outer products cancel in $(a+b)(a-b)$ because they are opposites, leaving $a^2 - b^2$ ; in $(a+b)(a+b)$ they are identical and combine into $2ab$ . And $73 \times 67 = (70+3)(70-3) = 4900 - 9 = 4891$ .

Distributing

(a+b)(a-b)

gives

a^2 - ab + ba - b^2

. Since

-ab

and

+ba

are opposites, they sum to zero and the middle vanishes. Distributing

(a+b)(a+b)

gives

a^2 + ab + ba + b^2

, where the two cross terms are the same sign and add to

2ab

. For the arithmetic, both

73

and

67

are

3

away from

70

, so set

a = 70

and

b = 3

: the product is

70^2 - 3^2 = 4900 - 9 = 4891

A student writes

(x + 7)^2 = x^2 + 49

. Identify the error, give the correct expansion, and show one numerical substitution that proves the student's version is wrong.

Answer: The student dropped the middle term $2(x)(7) = 14x$ ; the correct expansion is $x^2 + 14x + 49$ . Substituting $x = 1$ gives $(1+7)^2 = 64$ for the true value but only $1 + 49 = 50$ for the student's version.

Squaring does not distribute across addition. Writing

(x+7)^2

(x+7)(x+7)

and distributing gives

x^2 + 7x + 7x + 49

, and the two cross products combine to

14x

. Testing a value is the fastest way to expose this error, and almost any input exposes it — the one value to avoid is

x = 0

, where both versions happen to give

49

: at

x = 1

the correct expression gives

64

while the incorrect one gives

50

, a gap of

14

, which is exactly the missing

14x

evaluated at

x = 1

FAQ

Do I have to memorize the special product formulas, or can I just use FOIL every time?: FOIL will always give you the right answer, so nothing is lost mathematically. But the patterns matter for recognition, not just speed. In the next lessons you factor expressions like $x^2 - 121$ and $4x^2 + 20x + 25$ , and to do that you have to spot the pattern in reverse. Students who only ever distribute tend to miss those structures. Learn the patterns, and use FOIL as a check when you are unsure.
Is $(a-b)^2$ the same as $(b-a)^2$ ?: Yes. Expanding $(b-a)^2$ gives $b^2 - 2ab + a^2$ , which is the same three terms as $a^2 - 2ab + b^2$ in a different order. This makes sense because $b - a$ is the opposite of $a - b$ , and squaring any number and its opposite gives the same result. Note that this does not hold without the square: $a - b$ and $b - a$ are not equal unless $a = b$ .
Why doesn't $x^2 + 25$ factor?: The difference of squares pattern requires subtraction, because that is what makes the cross terms cancel. If you try $(x+5)(x+5)$ you get $x^2 + 10x + 25$ , and $(x+5)(x-5)$ gives $x^2 - 25$ . No pair of real binomials multiplies to $x^2 + 25$ . A sum of two squares is prime over the real numbers, and recognizing that quickly is useful in the factoring lessons ahead.
When can I use the mental math shortcut for multiplying two numbers?: Use the difference of squares whenever the two numbers are the same distance above and below an easy anchor, usually a multiple of ten or a hundred: $58 \times 62$ works because both are $2$ from $60$ . Use the squaring pattern whenever you need the square of a number close to a round one, like $103^2 = 10000 + 600 + 9$ . If the two factors are not symmetric about a round number, the difference of squares will not apply and you should multiply normally.

Learn this with a teacher, not a page

The Crimsora tutor teaches Special Products live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

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