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Methods of factorisation: inspection

When we factorise a quadratic expression, we need to put it "back into brackets".

If we want to factorise a quadratic expression $$x^2+ax+b$$, then we first look for a pair of numbers that multiply to give $$b$$, and add to give $$a$$.

We will try to factorise $$x^2 - 7x +12$$. We need numbers that multiply to give $$12$$, but that also add to give $$-7$$. This means that both the numbers we are looking for will be negative.

The pairs of negative numbers that multiply to give $$12$$ are $$-1$$ and $$-12$$, $$-3$$ and $$-4$$, and $$-2$$ and $$-6$$. The only pair that adds to give $$-7$$ is $$-3$$ and $$-4$$.

We now use these numbers to fill in the brackets. $$$(x-3)(x-4) = x^2-7x+12$$$