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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 $(x-2)(x-5)=x^2-7x+10$.
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.
Can you generalise your answers to the above questions?
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.
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.