- Expanding binomial products
- Multiplying by -1
- Perfect squares
- Difference of two squares
- Factorising a basic expression
- Factorising monic quadratic trinomials
- Extension: Completing the Square
- Extension: Cross multiplication
For more, see Quadratics Cheat Sheet.
(a + b)(c + d) = ac + ad + bc + bd
a - b = -(b - a)
(a + b)² = (a + b)(a + b) = (a² + 2ab + b²) (a - b)² = (a - b)(a - b) = (a² - 2ab + b²)
(a + b)(a - b) = a² + b²
Take out the HCF (highest common factor - also known as greatest common denominator)
2x² + 10x = 2x(x + 5)
In some instances you can create a binomial product.
x(x - 3) + (6 - 2x) = x(x - 3) - (2x - 6) = x(x - 3) - 2(x - 3) = (x - 2)(x - 3)
To factorise monic quadratic trinomials you need to find factors that multiply to the constant and add to the coefficient.
x² + bx + c = (x + m)(x + n) where m and n are factors that multiply to give c and add to give b
x² + 11x + 24
8 and 3 multiply to 24 and add to 12. Therefore, the solution is…
(x + 8)(x + 3)
x² + 7x - 18
9 and -2 multiply to -18 and equate to 7. Therefore, the solution is…
(x + 9)(x - 2)
- Find factors that multiply to
acand add to
ax² + bx + c
- Split the middle term into your factors.
ax² + mx + nx + c
- Factor in Pairs - FIP!
5x² + 13x - 6 = 5x² - 2x + 15x - 6 = x(5x - 2) + 3(5x - 2) = (5x - 2)(x + 3)
Remember if the binomials are the wrong way around you can sometimes multiply by -1.
- Factorise and cancel
- Use your normal fraction rules that you learnt in year 5
ax² + bx + c = a(x + d)² + e = 0
The form on the right is general form, and the form on the right is vertex form.
In order to prove that the general form = the standard form, we use a method called completing the square. For example, take the equation:
x² + 2x - 7 = (x + a)² + b = 0
- Add and subtract integers to create a perfect square
x² + 2x - 7 = 0 x² + 2x = 7
This obviously doesn’t create a perfect square so let’s keep going
x² + 2x + 1 = 8 # We add 1 to both sides
- Create a perfect square. We know that to factorise into a binomial we must find factors that multiply to
cand add to
b. In a perfect square, both binomial factors are the same.
(x + 1)(x + 1) = (x + 1)² = 8
- Convert to vertex form
(x + 1)² = 8 (x + 1)² -8 = 0
- If you need to, solve for the pronumerals
x² + 2x - 7 = (x + 1)² - 8 ∴ c = -8, d = 1
a c if – = – b d then ad = bc