You may also like

Powerful Quadratics

This comes in two parts, with the first being less fiendish than the second. It’s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.

Discriminating

You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.

Which Quadratic?

In this activity you will need to work in a group to connect different representations of quadratics.

Factorisable Quadratics

Age 16 to 18

This resource is from Underground Mathematics.
 


Factorisable quadratics

  1. The quadratic $x^2+4x+3$ factorises as $(x+1)(x+3)$. In both the original quadratic and the factorised form, all of the coefficients are integers.

    The quadratic $x^2-4x+3=(x-1)(x-3)$ similarly factorises with all of the coefficients being integers.

    How many quadratics of the following forms factorise with integer coefficients? Here, $b$ is allowed to be any integer (positive, negative or zero). For example, in part a, $b$ could be $-7$, since $(x-2)(x-5)=x^2-7x+10$.

    a.  $x^2+bx+10$
    b.  $x^2+bx+30$
    c.  $x^2+bx-8$
    d.  $x^2+bx-16$
    e.  $2x^2+bx+6$
    f.  $6x^2+bx-20$
  2. This time, it is the constant which is allowed to vary.

    How many quadratics of the following forms factorise with integer coefficients? Here, $c$ is allowed to be any positive integer.

    a.  $x^2+6x+c$
    b.  $x^2-10x+c$
    c.  $3x^2+5x+c$
    d.  $10x^2-6x+c$
  3. What are the answers to question 2 if $c$ is only allowed to be a negative integer?

Generalising

Can you generalise your answers to the above questions?

a.  Generalising question 1, if $c$ is a fixed integer, how many quadratics of the form $x^2+bx+c$ factorise with integer coefficients? Here, $b$ is allowed to be any integer.

b.  Further generalising question 1, if $a$ and $c$ are fixed integers, with $a$ positive, how many quadratics of the form $ax^2+bx+c$ factorise with integer coefficients? Again, $b$ is allowed to be any integer.
 
c.  How can we generalise question 2 or 3?
 
This is an Underground Mathematics resource.

Underground Mathematics is funded by a grant from the UK Department for Education and provides free web-based resources that support the teaching and learning of post-16 mathematics. It started in 2012 as the Cambridge Mathematics Education Project (CMEP).

Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.