Package of problems

## Suggestion

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. $x^2+bx+10$

What happens when you multiply out a pair of brackets?

Alternatively, if we knew the value of $b$, how would we go about attempting to factorise the quadratic?

How might this help us to work out which values of $b$ allow us to factorise the quadratic?

If you need a reminder of how to factorise quadratics where the $x^2$ term has a coefficient other than $1$, you may wish to look at the Quadratic grids resource.