Square of a Sum

Algebraic identities · ages 15–16

Cheat sheet

All the formulas in one place. Each one comes with an example and a button to practise it.

Square of a sum
(a + b)2 = a2 + 2ab + b2
a² ab ab b² a b a b

A square with side a + b is made up of a square a², a square b² and two rectangles ab. That is where the middle term 2ab comes from.

The first term squared, plus twice the product of the first and the second, plus the second term squared.

Example (x + 3)2 = x2 + 2·x·3 + 32 = x2 + 6x + 9 (2x + 5y)2 = 4x2 + 20xy + 25y2
Square of a difference
(a − b)2 = a2 − 2ab + b2

Just like the square of a sum, except that the middle term has a minus. The last term b² is always positive.

Example (2x − 5)2 = (2x)2 − 2·2x·5 + 52 = 4x2 − 20x + 25
Difference of squares
a2 − b2 = (a − b)(a + b)

It works both ways: it lets you multiply out brackets quickly and factorise expressions.

Examples 9x2 − 49 = (3x − 7)(3x + 7) 98 · 102 = (100 − 2)(100 + 2) = 10,000 − 4 = 9,996
Rationalising the denominator
a√m − n = a(√m + n)m − n2

Multiply the numerator and the denominator by the conjugate (the same binomial with the opposite sign). The denominator becomes a difference of squares and the root disappears.

Example 4√5 − 1 = 4(√5 + 1)5 − 1 = √5 + 1
Rearranging formulas

To make a variable the subject, do the same inverse operation to both sides: addition ↔ subtraction, multiplication ↔ division, squaring ↔ square root. Write the operation after a slash, as you would in your exercise book.

Example: make h the subject
A = a·h2/ ·2
2A = a·h/ ÷a
h = 2Aa
Common mistakes
(a + b)2 = a2 + b2(a + b)2 = a2 + 2ab + b2
(3x)2 = 3x2(3x)2 = 9x2
5 − (x − 2) = 5 − x − 25 − (x − 2) = 5 − x + 2
√a2 + b2 = a + b√a2 + b2 cannot be simplified
(√3)2 = 9(√3)2 = 3,  (2√3)2 = 4·3 = 12

Good to know: (a − b)2 = (b − a)2, because opposite numbers have the same square.

Extension

Cubesextension
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a − b)3 = a3 − 3a2b + 3ab2 − b3
a3 + b3 = (a + b)(a2 − ab + b2)
a3 − b3 = (a − b)(a2 + ab + b2)

In the sum and difference of cubes, the second bracket has a single ab (not 2ab), with the opposite sign to the one in the first bracket.

Example x3 − 8 = (x − 2)(x2 + 2x + 4)