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This comes in two parts, with the first being less fiendish than the second. Itâ€™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.

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

##### Age 16 to 18

This resource is from Underground Mathematics.

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 hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.

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.