Factorisable Quadratics
Factorisable quadratics
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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 $(x-2)(x-5)=x^2-7x+10$.
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. $6x^2+bx-20$ -
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. $10x^2-6x+c$ - 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?
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.
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