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SOLUTION

We seek to express this in the form (x + α)(x + β). To find the values of a and b, we solve, by

inspection, α + β = 7 and αβ = 12. Clearly the solutions are 4 and 3 (in either order), and no
other numbers satisfy these equations. Hence

x + 7x + 12 = (x + 4)(x + 3).
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Note that the order in which we write the brackets is unimportant.

Also note that the difference of squares factorisation could also be done using this method.
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For example, to factor x − 16, we solve α + β = 0 and αβ = −16. The solutions are 4 and −4
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work, so x − 16 = (x − 4)(x + 4).

This is, however, not a good method to use. It is better for students to be on the look out for
the difference of squares identity and apply it directly.

Students will need a lot of practice with factoring quadratics. It is worth mentioning here that in
further mathematics, both in the senior years and all the way through tertiary level
mathematics, quadratic expressions routinely appear and so being able to quickly factor them
is a basic skill.

QUADRATICS WITH COMMON FACTORS

We should always be on the look out for common factors before using other

factoring techniques.

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For example, to factor 3x + 9x + 6, we begin by taking out the common factor 3. We can then
proceed to factor further. Thus,

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3x + 9x + 6 = 3(x + 3x + 2)
= 3(x + 2)(x + 1)
Here we used the fourth identity after taking 3 common as a common factor.

Useful Link: Video- https://www.youtube.com/watch?v=CLxrOs_79iI