There are 3 standard ways of solving quadratic equations once they are in the form:
ax² + bx + c = 0
- Complete the square
- Use the formula
I think I generally teach them in that order probably without much thought as to why. I guess the formula needs to be derived by using completing the square and factorising seems to follow on from multiplying out double brackets, which comes before all of this. The I question that I sometimes get from students is “what’s the point in learning factorising if the two other methods always work?”. Well, it’s quicker and you can do it on a non-calc exam is probably a standard response.
But have you tried using the formula without a calculator to solve a quadratic that you know will factorise? Have a go. Plug this:
x² - 3x - 28 = 0
…and solve without a calculator.
It’s a surprisingly satisfying experience, one that I would not want to deny my students.
You’ll need to know your square numbers because b² – 4ac will always give you a square number for quadratics that factorise. But the arithmetic is perfectly reasonable and is likely to be so for most quadratics that can be factorised.
When I did this recently, I had a great question from one of my students, as I was aching my brain trying to make up a quadratic that I knew would factorise. “If you just picked one randomly, what are the chances that you would be able to factorise it?”
I’ve since had chance to investigate this further. It’s a great question and there is a lesson in here, or at least an extension question to explore once the fundamentals of using the formula are secure.
I started approaching it by using the formula and focussing on b² – 4ac and what values of a, b and c would yield square numbers. To simplify the problem, I started with a=1, so I was looking for when b² – 4c = 1,4,9,16,25, etc.
I then looked at it from the other end, i.e. starting with e.g. (x+1)(x+n) what values of a, b, and c are yielded. Then vary further to look at (x+m)(x+n). I just started with a few values of n and m to see if I could spot patterns. I won’t spoil the fun by revealing those patterns, but this is very open-ended and could provide some intrigue for the right learners.