Quadratic solving sorter

Quadratic solving sorter

Excerpt: Students are often unclear as to which is the most appropriate method for solving a particular quadratic equation. In this resource they sort a set of equations according to how they might solve them.

Problem

Sort these quadratic equations into groups.

• How will you choose to define the groups?
• Are there some equations which belong to more than one group?

Once you have sorted the equations into groups, write a further equation for each group. (There are blank cards provided for this purpose.)

How else could you have sorted the equations?

(For a suggestion as to one possible way of grouping the equations, you could use the possible grouping cards.)

You might want to download cards with the equations on them, so that you can sort them into groups more easily.

$3x^2-17=0$

$x^2-6x-10=0$

$4x^2-7x=0$

$3x^2-5x-11=0$

$x^2-3x+2=0$

$x^2+3x=0$

$(2x-5)^2-9=0$

$(2x-1)(x+3)=0$

$(x-3)^2-4=0$

$x^2-9=0$

$2x^2+5x+2=0$

$2x^2+7x+4=0$

$x^2+6x-72=0$

$(x+1)^2=0$

$x^2-16x=0$

$2x^2+6x+4=0$

$7x^2+14x-21=0$

$3(x+4)^2-8=0$

One possible grouping

Sort these quadratic equations into groups.

• How will you choose to define the groups?
• Are there some equations which belong to more than one group?

Once you have sorted the equations into groups, write a further equation for each group.

How else could you have sorted the equations?

Here is one possible way to group the equations. We ended up using seven groups. We felt that some of the equations fitted into more than one group, so we’ve drawn it like that.

What rules might we have used for our groups?

Here are the rules we used for our groups (and are those which appear on the possible grouping cards). They are all about how we would solve the equations.

Which rule goes with which group?

• Divide both sides by a common factor, then factorise.
• Rearrange if necessary, then take square roots of both sides. Rearrange to find $x$.
• Take out a factor of $x$, then either $x=0$ or the bracket equals zero.
• One of the brackets must be zero, so solve each one separately.
• Use completing the square or the quadratic formula to solve this equation.
• Use difference of two squares to factorise, then solve.
• Factorise into two brackets, then solve.

Our groups

• How does our grouping compare with yours?
• Which equations can only be solved in the way we’ve stated?
• Which equations could reasonably be solved in other ways?
• Are there any equations that could be solved in other ways, but it wouldn’t make sense to do so?
• Are there some equations which you would solve in one way if you had a calculator, but in a different way if you did not?
• Do you actually need to solve the equations in order to sort them?