The Quadratic Formula and Completing the Square: A Fresh LookAuthor(s): Mike SeagerSource: Mathematics in School, Vol. 36, No. 2 (Mar., 2007), p. 2Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215997 .

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The uadratic Formula and Completing the Squar

A FRESH LOOK by Mike Seager

I have been teaching for 15 years but I have always struggled to teach successfully the quadratic formula to my Higher Tier GCSE students. I want to show them where it comes from and not just present it as a piece of 'maths magic'; however, whenever I have approached the derivation via completing the square I know that I have left a lot of my pupils behind. This is especially so when we consider that, at GCSE level, they will only have to complete the square on expressions of the type x2 + bx + c but we have to help them through the algebraic manipulation for completing the square on ax2 + bx + c when deriving the quadratic formula.

This year, I think I solved the problem! Five minutes before the lesson (knowing that I was going to be observed by a visiting trainee teacher) I came to the realization that I could work backwards from the formula to the quadratic expression. A quick check of the algebra involved convinced me that this was the way to proceed:

-b + a b2 4ac X x=

2a

2ax = -b a b2_4ac

2ax + b = + b2-4ac

(2ax + b)2 = b2 -4ac

4a2x2 + 4abx + b2 = b2 - 4ac

4a2 X2 + 4abx + 4ac = 0

4a(ax2 + bx + c) = 0

ax2 +bx + c = 0.

(multiply by 2a)

(add b)

(square both sides)

(multiply out the bracket)

(collect like terms)

(factorize)

It may not be quite as mathematically sound and may still leave the pupils with some questions, but I felt that the pay- off in understanding the proof was worth it. Maybe everybody does it this way and I am the last one to realize!

I started the lesson by putting up three standard quadratics to solve and asked the class for the method. They soon solved the first two by factorizing but could not do the third: the class had already asked about this situation when we had

covered factorizing, so it was a fairly easy step to tell them that we were going to use a formula to solve quadratics of this type.

However, this did create a slight problem with completing the square, as I have always justified its inclusion at GCSE level (at least to myself) as a lead-in for the quadratic formula. On reflection though, I realize that I can concentrate now on linking y = x2 + bx + c with graph transformations by changing it to the form y = (x + d)2 + e. As graph transformations are the next topic on my scheme, it all seems to be working out beautifully.

As a last comment, I would also like to add that now I am teaching completing the square to my GCSE students by 'equating coefficients'. This has come from looking at the form of questions on the Higher Tier papers, which typically ask them:

Given that X2 + 2x + 5 - (x + a)2 + b find the values ofa and b.

By asking them to multiply out and equate, I have found that some of my weaker students are able to tackle this difficult question quite successfully. It also means that if the form of the expression changes slightly, then they have a method that will deal with that. I can then teach a more efficient method at AS level. La

Keywords: Quadratic formula; Completing the square.

Author Mike Seager, Chelmsford County High School for Girls, Broomfield Road, Chelmsford, Essex CM1 1RW. e-mail: mseager@cchs.essex.sch.uk

2 Mathematics in School, March 2007 The MA web site www.m-a.org.uk

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