Quadratic Factorisation

This section will explain how to factorise the quadratic equation with form y=x²+bc+c to the factorised form y=(x+n)(x+m), where n and m are integers. Factorisation is the reverse of expanding brackets, and after expanding the brackets the constants a, b, and c can be identified. Note that for this section a=1always; later sections will consider the case for general a.

Expand the following quadratic equation to determine the constants a and b.

The factorised form y=(x+n)(x+m) is considered for randomly chosen values of m and n:

The brackets (x+n) and (x+m) can be thought of as the lengths of the sides of a rectangle and the area of the rectangle is then y; this is shown below.

The rectangle has been divide into 4 sub-rectangles and you can press the buttons to see which rectangles contribute to a, b, and c constants, in the equation y=ax²+bx+c.

Using the table, can you spot a relation between m and n, and the constants b and c? I am going to break from the usual routine on this website and not reveal the pattern immeadiately. I recall, many classes in the past being silent in thought when presented with the table, and after 20 minutes the vast majority of students saw the pattern for themselves.

The next exercise requires thinking in reverse: you are given a,b, and c, and from this you must determine the constants m and n. Press the button to show the working out, however you are encouraged to try the exercise a few times before looking at the solution.

Summary

Factorisation of a quadratic equation to the form y=(x+n)(x+m) requires familiarity with the timestable.