I have found the topic of solving quadratic equations to be a bit of a funny one with students. In my experience, students are not too bad when presented with a quadratic expression and told to factorise it. However, when they are given a factorisable (pretty sure that's a word, if not it should be) quadratic as part of an equation, even when it is already nicely equal to zero, it can go horribly wrong, with mathematical laws broken left, right and centre.
The question above is designed to dig a bit deeper into the minds of students. Here is my thinking behind the choice of answers:
Answer A) Students go into "solving equations mode" and try to get the x on its own on the left-hand side. Mathematically, of course, there is nothing wrong with this, and students should be encouraged to continue to see where it gets them.
Answer B) Although this answer looks a bit weird, it is quite a common one in my experience, and it often occurs after students have dabbled fruitlessly with the situation left by starting with option A for a while. They may divide though by x correctly at first, but not liking the look of the nasty fraction, conveniently decide to forget to divide the 40 by x to leave a lovely looking linear equation.
Answer C) This is an incorrect factorisation. I have found that when students first learn to combine factorising with solving quadratic equations, they can begin to get a little mixed up with the signs, despite being okay when asked to simply factorise. In my opinion, this is not helped by the all too common teaching practise when solving equations of telling students to "swap the signs in the brackets to get your final answer". If students do choose this option, it is worth digging deeper to see what their reasons are.
Answer D) is correct
In the (hopefully near) future, we will have the facility on the website to not only see how students from all over the world attempt questions like this, but also to read their explanations for their answers. I cannot wait for this.
In the meantime, try out the question with your students.
- Do they fall into the traps set for them by the incorrect answers?
- Can they explain why each of the wrong answers have been chosen?
- Can they think of another incorrect answer and explain their reasons for it?
- Can they design a diagnostic question of their own to test their classmates' abilities to solve quadratic equations?