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

              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.

Max V. A Level French tutor, A Level Chemistry tutor, A Level Geograp...

1 year ago

Answered by Max, who has applied to tutor A Level Maths with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Liam O.

Degree: Economics (Bachelors) - Cardiff University

Subjects offered: Maths


“About me: My name is Liam, I study Economics at Cardiff business school. I have always thought of myself as a patient, understanding person who is always keen for the opportunity to help others. I have previously tutored reading at pr...”

MyTutor guarantee

£20 /hr

Harry R.

Degree: Mathematics and Computer Science (Masters) - Durham University

Subjects offered: Maths, Physics+ 1 more


“About Me: I am a Maths and Computer Science student at Durham University and I have always enjoyed problem-solving and the reward of solving a problem that I perhaps didn't think I could. I am very patient and friendly. I have volunte...”

MyTutor guarantee

£20 /hr

James B.

Degree: Mathematics (Masters) - Exeter University

Subjects offered: Maths


“About MeI'm a second year student at Exeter University. I've been passionate about maths from a young age and have carried that love through my school and uni career. I hope that I can generate the same passion for my tutees as well....”

About the author

£20 /hr

Max V.

Degree: Geology (Bsc) Hons (Bachelors) - Edinburgh University

Subjects offered: Maths, Physics+ 6 more

Further Mathematics

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

Find the stationary points of the function y = (1/3)x^3 + (1/2)x^2 - 6x + 15

[FP2] Solve: 3 cosh x - 4 sinh x = 7

The point P lies on the curve C: y=f(x) where f(x)=x^3-2x^2+6x-12 and has x coordinate 1. Find the equation of the line normal to C which passes through P.

Differentiate a^x

View A Level Maths tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss