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 \((x-2)(x-5)=x^2-7x+10\).
\(x^2+bx+10\)
\(x^2+bx+30\)
\(x^2+bx-8\)
\(x^2+bx-16\)
\(2x^2+bx+6\)
\(6x^2+bx-20\)
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.
\(x^2+6x+c\)
\(x^2-10x+c\)
\(3x^2+5x+c\)
\(10x^2-6x+c\)
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?
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.
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.
How can we generalise questions 2 and 3?
