What is a quadratic trinomial?
A trinomial has three terms. A quadratic has x² as its highest power. So x² + 5x + 6 is a quadratic trinomial.
Its general form is ax² + bx + c. Here a is the number before x², b is the number before x, and c is the number on its own.
To factorise means to write the expression as a product of simpler brackets. It is the opposite of expanding.
See it with algebra tiles
A big square tile has area x². A strip has area x. A small square has area 1. For x² + 5x + 6 you take 1 big square, 5 strips and 6 small squares.
Join them into one rectangle. The width is x + 3 and the height is x + 2. The area of a rectangle is width times height, so
x² + 5x + 6 = (x + 2)(x + 3)
If your tiles cannot make a rectangle, the trinomial cannot be factorised with whole numbers.
Factorising x² + bx + c
Expand (x + m)(x + n) and you get x² + (m + n)x + mn. So we need two numbers m and n where m × n = c and m + n = b.
- List the pairs of numbers that multiply to c.
- Pick the pair that adds to b.
- Write (x + m)(x + n).
- Expand to check.
Example: x² + 7x + 12. Pairs for 12: 1 and 12, 2 and 6, 3 and 4. Only 3 + 4 = 7. So (x + 3)(x + 4).
Signs
| c | b | The two numbers | Example |
|---|---|---|---|
| + | + | both + | x² + 7x + 12 = (x + 3)(x + 4) |
| + | − | both − | x² − 7x + 12 = (x − 3)(x − 4) |
| − | + | one + and one −; the bigger one is + | x² + 2x − 15 = (x + 5)(x − 3) |
| − | − | one + and one −; the bigger one is − | x² − 2x − 15 = (x − 5)(x + 3) |
If c is positive, both numbers have the same sign. If c is negative, they have different signs.
Factorising ax² + bx + c (split the middle term)
When a is not 1, multiply a and c. Find two numbers that multiply to a × c and add to b. Use them to split the middle term. Then group.
Example: 2x² + 7x + 3. a × c = 6. The numbers 6 and 1 multiply to 6 and add to 7.
2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1).
The tiles agree: 2 big squares, 7 strips and 3 small squares make a rectangle with sides x + 3 and 2x + 1.
Special trinomials
Perfect squares: x² + 2mx + m² = (x + m)² and x² − 2mx + m² = (x − m)². Example: x² + 10x + 25 = (x + 5)². The tiles make a square.
Difference of squares: x² − m² = (x − m)(x + m). It has no middle term, because the +m and −m cancel.
Always check: expand your brackets. You must get back the original expression.
Try it: a practical
In the 3D, set a = 1 and move p and q. Before you look, predict the trinomial. Then read it from the screen. At home, cut a square of paper for x², and strips and small squares from card. Build x² + 4x + 3 into a rectangle and read its sides.
Key formulas and definitions
- (x + m)(x + n) = x² + (m + n)x + mn
- x² + bx + c: find m, n with m × n = c and m + n = b
- ax² + bx + c: find m, n with m × n = a × c and m + n = b, then split and group
- x² ± 2mx + m² = (x ± m)²
- x² − m² = (x − m)(x + m)
Worked examples
1. Factorise x² + 7x + 12.
We need two numbers that multiply to 12 and add to 7. They are 3 and 4. So x² + 7x + 12 = (x + 3)(x + 4).
2. Factorise x² − 8x + 15.
c is positive and b is negative, so both numbers are negative. Multiply to 15 and add to −8: −3 and −5. So (x − 3)(x − 5).
3. Factorise x² + 3x − 10.
c is negative, so one number is + and one is −. Multiply to −10, add to 3: +5 and −2. So (x + 5)(x − 2).
4. Factorise x² − x − 20.
Multiply to −20, add to −1: −5 and +4. So (x − 5)(x + 4).
5. Factorise 2x² + 7x + 3.
a × c = 6. Two numbers that multiply to 6 and add to 7: 6 and 1. 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1).
6. Factorise 6x² − 5x − 6.
a × c = −36. Two numbers that multiply to −36 and add to −5: −9 and +4. 6x² − 9x + 4x − 6 = 3x(2x − 3) + 2(2x − 3) = (2x − 3)(3x + 2).
7. Factorise x² + 10x + 25.
25 = 5 × 5 and 5 + 5 = 10. So x² + 10x + 25 = (x + 5)(x + 5) = (x + 5)².
8. Solve x² − 5x + 6 = 0.
Factorise: (x − 2)(x − 3) = 0. A product is zero only if a factor is zero, so x = 2 or x = 3.
Common mistakes
- Choosing numbers that multiply to c but forgetting to check that they add to b.
- Getting the signs wrong. If c is negative, the two numbers must have different signs.
- Forgetting to multiply a and c when a is not 1.
- Not expanding at the end to check the answer.