Algebraic manipulation.
Expand to reveal terms. Factorise to reveal structure. Complete the square to reveal a turning point.
Collect like terms · expand multiple brackets · factorise fully · complete the square
Collect like terms and expand
Three apples plus two apples make five apples, but three apples plus two oranges cannot become five apples. Collecting algebraic terms works the same way: the letter part must match before their coefficients can be combined.
Brackets describe a whole group. Multiplying that group by 3 means multiplying every term inside it, not just the first. Begin with small numerical groups to understand this distributive idea, then use it when the terms contain letters.
Like terms have exactly the same letters raised to the same powers. Combine their coefficients; keep the letter part unchanged.
Simplify 2a² + 3ab − 1 + 5a² − 9ab + 4.
= 7a² − 6ab + 3
Expand 3x(2x − 4y) and (3x + y)(x − 4y).
Distribute each term to every term in the other factor.
(3x + y)(x − 4y) = 3x² − 12xy + xy − 4y²
= 3x² − 11xy − 4y²
Expand (x − 2)(x + 3)(2x + 1).
First two brackets: x² + x − 6.
= 2x³ + 3x² − 11x − 6
Pause and explainWhy is 3x + 2x equal to 5x, but 3x + 2 not equal to 5x?
The first expression counts five lots of x. In the second, the 2 is a separate constant, not two lots of x.
Factorise fully: look for structure
Expanding turns a product into a sum. Factorising reverses that direction: it rewrites a sum as a multiplication. To find a common factor, ask what every term was multiplied by.
For 6x + 9, both terms contain a factor of 3, so write 3(2x + 3). The terms left inside the bracket are the results of dividing each original term by 3. Expanding your answer is a direct check that nothing was lost.
Factorising reverses expansion. Start by taking out the highest common factor, including common letters.
Factorise 9x² + 15xy.
Expanding checks the answer. Taking out only 3 would leave a common x inside, so the factorisation would be incomplete.
| Pattern | Factorisation |
|---|---|
| ax + bx + kay + kby | x(a + b) + ky(a + b) = (a + b)(x + ky) |
| a²x² − b²y² | (ax − by)(ax + by) |
| a² + 2ab + b² | (a + b)² |
| a² − 2ab + b² | (a − b)² |
Factorise 6x + 9 + 4xy + 6y.
= (2x + 3)(3 + 2y)
Pause and explainHow can you check that 4(x + 2) is a factorisation of 4x + 8?
Expand it: 4 × x + 4 × 2 gives 4x + 8, which matches the original.
Quadratics and cubics
When two brackets are multiplied, every term in one must multiply every term in the other. The middle terms of a quadratic often come from two separate products, which is why their sum matters.
Factorising a quadratic reverses those products. For a monic quadratic, seek two numbers with the required product and sum. In a cubic, first look for a common factor or an available linear factor so that the remaining problem becomes a quadratic you already recognise.
Factorise 6x² + 11x + 3.
- Multiply first and last coefficients: 6 × 3 = 18.
- Find two numbers with product 18 and sum 11: 9 and 2.
- Split the middle: 6x² + 9x + 2x + 3.
- Group: 3x(2x + 3) + (2x + 3) = (3x + 1)(2x + 3).
Factorise 2x³ + 2x² − 12x.
Take out 2x first, then factorise the quadratic.
Pause and explainWhy does (x + 2)(x + 3) contain a 5x term?
Its middle products are 3x and 2x. They collect to 5x after all four products are written.
Complete the square
A perfect-square bracket has a predictable expansion: (x + a)² = x² + 2ax + a². Completing the square rewrites a quadratic so that this pattern becomes visible.
Take half the coefficient of x to choose a. Squaring that bracket introduces a constant, so compensate by subtracting the same amount. The expression’s value must stay unchanged. This form later reveals a quadratic graph’s turning point and can help solve equations.
For x² + bx, halve b to build a square, then compensate for the constant introduced.
Complete the square for x² + 6x + 2.
= (x + 3)² − 7
Expanding (x + 3)² gives x² + 6x + 9, so subtract 7 to obtain the original +2.
Complete the square for 2x² − 12x + 5.
= 2[(x − 3)² − 9] + 5
= 2(x − 3)² − 13
Factor out the coefficient of x² from the first two terms first. The minimum is −13, at x = 3.
Pause and explainWhy does x² + 6x become (x + 3)² − 9?
The square expands to x² + 6x + 9. Subtracting 9 removes the extra constant.
Your turn. Expand to check.
Choose the direction of the task before starting: expand products, collect like terms, or factorise a sum. A question asking for factorisation is not finished with an expanded expression.
Work one bracket at a time and show negative signs explicitly. Expand a proposed factorisation or completed square afterward; recovering the original expression is a reliable structural check.
Expand (x − 1)(x + 2)(x + 3).
Reveal worked answer
Factorise fully 12x³ − 27x.
Reveal worked answer
Factorise 2x² − 7x + 3.
Reveal worked answer
Write 3x² + 12x + 7 in completed-square form.