Seeing Structure in Expressions
Algebraic identities
Use (a + b)², (a - b)² and a² - b² to expand brackets and to multiply quickly.
An identity is not an equation waiting to be solved. It is a statement that stays true no matter what numbers you put in. Once you trust one, you can use it in both directions: to open brackets out, and to fold a messy expression back up.
The three to know
(a + b)² = a² + 2ab + b². (a - b)² = a² - 2ab + b². (a + b)(a - b) = a² - b². Everything in this skill is one of those three, wearing a disguise.
Worked example
Why is (a + b)² not simply a² + b²?
(a + b)² means (a + b)(a + b), so multiply every part of the first bracket by every part of the second.
Squaring a bracket squares the whole thing, not each piece separately.
Try it together
Now use an identity to work out 102² in your head.
102 is awkward, but 100 is not. Write 102 as 100 + 2 and use (a + b)² with a = 100 and b = 2.
1.First term: what is a², that is 100²?
The difference of two squares
a² - b² = (a + b)(a - b) turns 105 × 95 into (100 + 5)(100 - 5), which is 10000 - 25 = 9975. Look for it whenever two numbers sit the same distance either side of a round one.
Have a go
Have a go on your own: use (100 + 1)² to work out 101².
Hint: 10000 + 200 + 1.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.