Factorising into Two Brackets when the leading term is x squared

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We may have to factorise expressions such as

When the term is exactly and not for example, this is very simple. We look for two numbers that add to give 7 and multiply to give 12. Two such numbers, the only two in fact are 3 and 4, hence the expression factorises as

The order in which we write the answer does not matter. We could also write

If we have no minus signs, it is always as simple. If there are minus sign, as in the expressions

the the above paragraph changes very slightly. “ We look for two numbers that add to give 7 and multiply to give 12” becomes We look for two numbers that add or subtractto give -2 and multiply to give 8”. Two such numbers, the only two in fact, are 2 and 4:

2-4=-2 and 2*4=8 hence the expression factorises as

Here is another example. Remember that two minuses multiply to make a plus.

The only two numbers that multiply to give 15 are 1, 15 or 3, 5 or -1, -15 or -3, -5. The only way we can make -8 from any of these pairs of numbers is -3 -5=-8. Hence the expression factorises as

Here are some more examples:

We could write each of these the other way round. The last one for example could be written,

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