Expanding Squared Brackets

Squaring a bracket does not mean squaring each term inside it. $(x+3)^2$ is not $x^2 + 9$ — a middle term appears, and forgetting it is the single most common error in algebra. Two identities cover every case.

Concept The two identities

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

In words: the square of the first, plus twice the product of the two, plus the square of the second.

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

The subtraction version differs in exactly one place — the middle term becomes negative. The first and last terms stay positive in both cases, because squaring any quantity gives a positive result.

Note Where the middle term comes from

A square of side

a+b

splits into four pieces:

a^2

b^2

, and two rectangles of area

ab

. Those two rectangles are the

2ab

that so often goes missing.

Example Both signs

Expand $(x+3)^2$ and $(x-5)^2$ .

For the first,

a = x

and

b = 3

(x+3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9

For the second,

a = x

and

b = 5

, with a minus sign.

(x-5)^2 = x^2 - 2(x)(5) + 5^2 = x^2 - 10x + 25

⟹ x² + 6x + 9 and x² − 10x + 25

Notice that $+25$ is positive even though the bracket contained a minus — $(-5)^2 = 25$ .

Example When the first term has a coefficient

Expand $(2x+1)^2$ and $(3x-2)^2$ .

Here

a = 2x

, so

a^2 = (2x)^2 = 4x^2

— the coefficient is squared too.

(2x)^2 + 2(2x)(1) + 1^2 = 4x^2 + 4x + 1

For the second,

a = 3x

and

b = 2

(3x)^2 - 2(3x)(2) + 2^2 = 9x^2 - 12x + 4

⟹ 4x² + 4x + 1 and 9x² − 12x + 4

Writing $(2x)^2$ as $2x^2$ instead of $4x^2$ is a frequent slip. The whole term is squared, coefficient included.

Note Watch the signs

With a plus inside the bracket, the middle term is positive.

With a minus inside the bracket, the middle term is negative.

The first and last terms are always positive, whichever sign appears.

That is the only difference between the two identities, so there is really just one pattern to learn.

Note Mistakes to avoid

Writing

(x+3)^2 = x^2 + 9

— the

6x

is missing.

Forgetting to double the product: the middle term is

2ab

, not

ab

Leaving a coefficient unsquared, as in

(2x)^2 = 2x^2

Making the last term negative after a subtraction;

(-b)^2

is always positive.

Applying these identities to

(a+b)(a-b)

, which is a different pattern altogether.

Summary

(a + b)² = a² + 2ab + b², and (a − b)² = a² − 2ab + b².
Only the middle term changes sign; the first and last terms are always positive.
The 2ab comes from the two identical rectangles inside the square — it is never optional.
Square the whole term, coefficient included: (2x)² = 4x², not 2x².
Squaring a bracket is not the same as squaring each term separately.