10 January 2009

Baddie of the Month – Factoring a Quadratic with a≠1

Filed under: applets,education,math,sloodle — Tags: , , , , , — admin @ 5:13 am

Goodie: “A technique/question that can be applied in many places and teaches thinking.”

Baddie: “A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

I am going to try to blog a baddie and a goodie per month. We shall see and of course – this is my opinion.

January 2009 Baddie of the MonthFactoring a quadratic with a≠1 “by hand”.

Okay, I can mostly understand learning to factor “by-hand”:   x²+3x+2  or   x²+x-2.  

Once you understand the principles and get the technique (my scheme), factoring a quadratic by hand with a=1 is faster than using the quadratic formula.

But, I absolutely and totally do not understand the reasoning behind other factoring-by-hand techniques!

Why not: Factoring techniques

  • serve no useful purpose – once factored, with a≠1 you must still solve the individual factors.
  • don’t always work – MOST quadratics even with a=1 and real roots CANNOT be factored by hand.
  • are hard to learn, there are many “special cases”, they take alot of time to teach, …

What to do and why: Use the quadratic formula for all your factoring needs.

  • We are going to teach them the quadratic formula anyway.
  • It always works – either we get real roots and can factor or we get non-real roots and know we cannot factor.
  • By using the quadratic formula all of the time, the connection between quadratics, roots, x-intercepts, graphs of quadratics becomes clear.
  • Repetition of a single technique is much more likely to stay in their heads.

Conclusion: Don’t teach factoring by hand except when a=1. Use the quadratic formula.

Here’s how: ax²+bx+c=a(x-x1)(x-x2) where x1=(-b+D))/2ax2=(-b-D))/2a,  D=√(b²-4ac)

(Here “by hand” means looking for the factors without a formula like when you say “The factors of 2 are 1 and 2 and oh yes, they add to 3 (first expression) or the factors of 2 are 1 and 2 and oh yes, they subtract to 1 (second expression)”.)

Related topics to come in future blogs

Please, OMG please don’t teach completing the square – an even worse waste of time than hand factoring.

Please, please don’t teach complex numbers in the same 2 month span as you teach graphing of quadratics.


  1. [...] Here is an interesting math blog: [...]

    Pingback by Associated Colleges of Illinois: Mentor Online » Blog Archive » More sites for math — 6 February 2009 @ 8:28 am

  2. [...] Past: January 2009 Baddie of the Month: Hand-factoring a quadratic with a≠1. [...]

    Pingback by Baddie of the Month - Teaching “completing the square” for quadratics « SL2RL-Math247 — 8 February 2009 @ 4:13 am

  3. [...] Past: January 2009 Baddie of the Month: Hand-factoring a quadratic with a≠1. [...]

    Pingback by Baddie of the Month - Teaching complex numbers with the quadratic formula « SL2RL-Math247 — 8 March 2009 @ 12:14 am

  4. A. HALIM…

    Although your article about math graphs sounds interesting but i’m not sure if i could agree with you in 100%….

    Trackback by A. HALIM — 19 March 2009 @ 8:21 pm

  5. Formula for 13x^2 + 14x + 1 ? No. Tsk tsk.

    If it takes less than a minute to check if an expression is factorable, why not? Before running the tedious formula.

    Special cases? I have none. One method for all, allows to check if an expression is factorable before doing the work.

    Try this: Teaching Factoring – Should we? and I’d be happy to point you towards a discussion of trinomial factoring by breaking the middle.

    Comment by Jonathan — 12 April 2009 @ 4:02 pm

  6. If only the problem was “Factor 13x^2 + 14x + 1″. My sample problems are: “Using either Lagrange’s or Newton’s interpolation method, find the polynomial of smallest order that passes through the points (-1,0), (0,1) and (1,28). Graph this function finding all intercepts and extreme values and show that it goes through these points.”. These kids don’t have time to mess around with factoring – which NO ONE ever remembers or uses after Algebra 1 or with completing the square. The quadratic formula is tedious? No way! It is an easy to remember formula that works every time.

    Comment by admin — 26 April 2009 @ 10:20 pm

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