Algebra

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Factorising Quadratic Expressions

We can factorise quadratic expressions into two sets of brackets. When we multiply out of these brackets, we will generate the original expression.

Quadratic expressions can also be factorised, but this time we need to factorise into 2 sets of brackets.

Let's factorise $x^2 + 5x + 6$

We want to achieve the form $(x+a)(x+b)$

Similar to before, the goal is to create two brackets such that when we multiply them together, we can regenerate the original expression.

First, write two brackets to create the $x^2$ term

$(x + ...)(x + ...)$ will multiply together to give an $x^2$

When we expand brackets, we multiply each term

Multiplying the first terms $({\color{#21affb}x}+a)(x+{\color{#21affb}b})$ and inside terms $(x+{\color{#21affb}a})({\color{#21affb}x}+b)$ creates two $x$ terms which we add together.

Multiplying the last terms creates a product

By multiplying the last terms $(x+{\color{#21affb}a})(x+{\color{#21affb}b})$ , we generate a number which is the product of the two numbers in the brackets.

To finish the factorisation, fill in the brackets

The general form of a quadratic is $ax^2+bx+c$ . We need to find two numbers which multiply to give $c$ and add together to give $a$ .

Which two numbers add to make 5 and multiply to make 6?

$2$ and $3$ add to 5 and multiply to make 6

These are the values we will put in our brackets. $(x+2)(x+3)$

To check, expand the brackets

$(x+2)(x+3)$ multiplies to give $x^2 + 5x + 6$ , which is our original equation! Great 😀

What is the factorisation of $x^2 -12x + 36$ ?