Package of problems

## Problem

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 (a), $b$ could be $-7$, since $(x-2)(x-5)=x^2-7x+10$.

1. $x^2+bx+10$

2. $x^2+bx+30$

3. $x^2+bx-8$

4. $x^2+bx-16$

5. $2x^2+bx+6$

6. $6x^2+bx-20$

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.

1. $x^2+6x+c$

2. $x^2-10x+c$

3. $3x^2+5x+c$

4. $10x^2-6x+c$

3. What are the answers to question 2 if $c$ is only allowed to be a negative integer?

### Generalising

1. 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.
2. 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.