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Completing the square – What more can you do?

In my last post, which was a while ago, I looked at the three different ways to solve a Quadratic Equation.  The last of these, ‘completing the square’ may bot seem as obvious as the others, but in fact that is how the formula is found. Also there are questions in GCSE exams, especially on the Higher papers, that do guide the student through this method.

What is useful in the Completing the Square’ method is it can help find minimum and maximum values. You will learn another way of doing that later, called Calculus.

To find the minimum value of x2 + 4x – 1 we can complete the square by adding in 5, but that 5 must then be taken away.

x2 + 4x – 1 = x2 + 4x – 1 + 5 – 5 = (x + 2)2 – 5
We know that (x + 2)2 will be 0 when x = -2, but can never have a lower value.

This means the lowest value of x2 + 4x – 1 is -5 and this is when x = -2.  We can also show that is true by looking at the graph of this function.


This graph has been created by this website. Its called Desmos and looks like a state of the art way of creating graphs.

Go have a look at it!  There are beautiful ‘sliders’ that help you see how graphs change when the equation does.



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