quadratic equation, completing the square

[mollykins]

yesterday in maths, we were going over, how to solve quadratic equations by completing the square, and the teacher put some equations to solve on the board.

He said that if we were confident and could easily do the rest, we could try the last one, which is the kind of thing you do at a-level.

However, i didn't have time to do it, but jotted down the question and the answer, but can't for the life of me work out how to get from one to the other, by completing the square.

so how do you turn ax² + bx + c = 0

into x = -b plus/minus the square root of b² - 4ac
............---------------------------------
-------------------------

....................................2a

????????? thaks for the help, i hope it turns out in the right format so you can udnerstand the equation.

[Bloss]

I always show Higher groups the derivation of the quadratic formula through completing the square, but not other groups unless they wish to see it.
I currently teach a GCSE course which includes differentiation and functions - makes teaching AS much easier..

[chakka35]

We did it at my Liverpool grammar school at fourth-form level, a year before taking School Certificate (a more challenging form of the later 'O'-level) and I surprised myself a month or so ago by remembering how to derive the formula by completing the square - that's over sixty years later.
This is not a boast about me, but about the standard of education available to a working-class lad in those days.

[jake-the-peg]

Just an an aside that you might find interesting.

There are Bablyonian cuniform tablets that talk of fields with a certain area and have one side 20 cubits (or whatever) longer than the shorter side . This is a quadratic equation when you multiply it out. In the real world - you just go and measure the field so this was clearly a maths problem solving a quadratic nearly 4,000 years ago

Third order (cubic) equations with (ax³+b²+cx+d) are a lot harder and were only properly solved in the 1500's in Italy.

There was a great rivaly between two mathematicians Cardano and Tartaglia over who was the first - but the point is that in order to do it they invented/discovered complex numbers (the square root of negative numbers)

This was a rather amazing feat in many ways but not least that this was probably the first mathematical discovery made in Europe rather than in the far East and in particular India

[factor-fiction]

I enjoyed reading that aside, jake-the-peg. Thanks.

You meant to type ax³+bx²+cx+d = 0 as your cubic equation.

There's a good intruductory article here on how to solve them:
http://www.dpmms.cam.ac.uk/~wtg10/cubic.html