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

You can try to guess the roots, but it 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:

Now subtract c/a from both sides:

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

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:

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:

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:

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

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

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

Now subtract b/2a from both sides: