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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.

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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.
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