A letter is a number you do not know yet
Algebra is arithmetic with some of the numbers not yet filled in. Every rule you already use still applies — the letter simply stands in until a value arrives.
Some vocabulary is examined directly. A term is a single item such as 5x². Its coefficient is the number in front, here 5. An expression is terms combined, like 5x² − 3x + 2. An equation has an equals sign and can be solved. A formula is an equation expressing a rule, such as A = πr². An identity is true for every value, and is written with ≡.
Expression, equation, formula, identity
3x + 2 is an expression — there is nothing to solve. 3x + 2 = 11 is an equation, true only when x = 3. v = u + at is a formula, a rule connecting quantities. 2(x + 3) ≡ 2x + 6 is an identity, true for every x. Questions asking you to "solve" an expression are testing whether you know the difference.
Collecting like terms
Terms can be added only if they are like — the same letters raised to the same powers. 3x and 5x combine to 8x. 3x and 5x² do not combine at all, because they are different things: three lengths and five areas cannot be added.
The coefficients add; the letter part is untouched. 3x + 5x is 8x, not 8x² — you are counting how many x there are, not multiplying them.
Simplify 4a + 3b − 2a + 5b² − b + 2b².
- Sort the terms by type: a terms, b terms, b² terms.Doing this on paper before combining anything prevents terms being lost.
- a terms:
4a − 2a = 2a. - b terms:
3b − b = 2b.A lone b has coefficient 1, which is easy to overlook. - b² terms:
5b² + 2b² = 7b².These cannot join the b terms — different power, different quantity. - Answer:
2a + 2b + 7b², conventionally written in descending powers:7b² + 2a + 2b.Either order is accepted; consistency matters more than the choice.
7b² + 2a + 2b
Expanding brackets
A bracket means everything inside is multiplied by whatever is outside — everything, not just the first term. 3(x + 4) is 3x + 12.
A minus sign in front changes the sign of every term inside: −2(x − 5) is −2x + 10. That second sign change is the single most common error in the topic.
For two brackets multiplied together, every term in the first must meet every term in the second — four products for two binomials.
Squaring a bracket
(x + 3)² means (x + 3)(x + 3), which expands to x² + 6x + 9. It does not mean x² + 9. Test it with a number: at x = 1 the correct expansion gives 16, and x² + 9 gives 10. Substituting a value is a five-second check that catches this every time.
Substitution and rearranging
Substitution replaces letters with numbers, and the order of operations still governs everything. Brackets are compulsory around a negative value: substituting x = −2 into 3x² gives 3 × (−2)² = 12, whereas writing 3 × −2² would give −12.
Rearranging a formula makes a different letter the subject, and it uses exactly the same moves as solving an equation: do the same thing to both sides until the wanted letter stands alone. The only difference is that the answer contains letters rather than a number.
Make r the subject of A = πr², and evaluate 3x² − 2xy when x = −3 and y = 4.
- Divide both sides by π:
A/π = r².Undo the multiplication first, working outwards from r. - Square-root both sides:
r = √(A/π).The positive root only, since r is a radius. - For the substitution, bracket the negative:
3(−3)² − 2(−3)(4).Without brackets the squaring would apply only to the 3. = 3(9) − (−24) = 27 + 24 = 51.Minus a negative is a plus — the second sign change is where marks are lost.
r = √(A/π); the expression evaluates to 51
Before you leave this chapter
- Expression, equation, formula, identity — four different things.
- Only like terms combine: 3x + 5x = 8x, but 3x + 5x² stays as it is.
- A bracket multiplies everything inside, and a minus outside flips every sign inside.
- (a + b)² = a² + 2ab + b². The middle term is not optional.
- Bracket negative values when substituting, and rearrange by doing the same to both sides.
Expanding brackets as an area
Multiplying two brackets produces four terms, and students who lose one usually lose the same one — the second cross term. Drawing the product as a rectangle makes it impossible to miss, because each of the four pieces is a visible region.
A rectangle of width x + a and height x + b splits into four smaller rectangles whose areas are x², ax, bx and ab. Add them and you have the expansion.
Switch to (x+a)². Both strips are the same size, so the middle term is exactly twice the number in the bracket — which is the term everyone forgets when squaring.
The check that takes five seconds
Substitute a small number into both the original and your expansion. For (x + 3)(x − 2) at x = 1: the original gives 4 × (−1) = −4, and the expansion x² + x − 6 gives 1 + 1 − 6 = −4 ✓. If the two disagree, the expansion is wrong, and you have found out in one line.