# How do I find the minimum or maximum of a quadratic function?

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Take a general function f(x) = ax²+bx+c

First thing to do with every problem is sit back and think a bit: there are usually several ways to solve a problem but one will likely be easier and shorter.

In this case we don’t want to start differentiating the function and looking for points where f’(x)=0 . This method does work, but it can involve quite a bit more fiddling around than other ways.

Since this is a quadratic this problem can be solved by simply ‘completing the square’ - we need to get our function into the form a(x+d)²+e . With this done we can simply read off the minimum or maximum e reached when x+d=0 (x=-d).

To get the quadratic into the right form we just need to follow a few simple steps :

1) Factor out a -> ax²+bx+c = a(x²+b/ax+c/a)

2) Add and subtract by a number d to such that the form ‘x² + 2dx + d²’ is present:

= a(x²+b/ax+(b/2a)²-(b/2a)²+c/a)

3) Rearrange the ‘x² + 2dx + d²’ form into ‘(x+d)²’

= a((x+b/2a)²-(b/2a)²+c/a)

4) Multiply out by a to get the final expression

= a(x+b/2a)²-a(b/2a)²+a(c/a)

= a(x+b/2a)²+(-b²/4a+c)

So here the minimum or maximum is given by (-b²/4a+c). Whether it is a maximum or minimum will depend on the sign of a.

As you can see the final expression gives us d= b/2a and e= (-b²/4a+c). These values are very quick and easy to derive so do not need to be learned by heart.

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