Quadratic Equations

This is for Thomas and Emma and all those who loved mathematics at school.

We want to solve a quadratic equation which can be written in the form

qe1

you can try to guess the roots but is very rarely easy outside of school text books. However we can solve it by a method called ‘completing the square’ for reasons that will become obvious later.

First of all we divide everything by a:

qe2

Now subtract c/a from both sides:

qe3

we can now factor out x from the left hand side:

qe4

We can think of this as a rectangle – one side length x and the other side length x+b/a. The area of this rectangle is x times x+b/a which is also equal to -c/a:

qe5

We can now cut the right hand rectangle (turquoise) in half so its width is now b/2a. Then we turn it round and moving it to the bottom of the red square:

qe6

We now have a square of area x+b/2a by x+b/2a.

We know the area of the coloured part it is -c/a. There is also an extra bit. The area of this extra bit is b/2a x b/2a = b2/4a2. We can write this down as a formula:

qe7

Now we can take the square root of both sides remembering that the square root can be plus or minus. Therefore we have:

qe8

Now we can simplify this equation by multiplying the top and bottom of c/a by 4a

qe9

Now take out the 4a2 from the square root (giving 2a):

qe11

Now subtract b/2a from both sides:

qe10

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