Hi all,

this is a question I've been doing. Let $\displaystyle n$ (known) be some composite odd integer. Find two numbers $\displaystyle b$ and $\displaystyle c$ such that :

- $\displaystyle x^2 + bx + c + n = 0$ is factorable over the real numbers, with integer coefficients in the standard form and factorized form.

- $\displaystyle x^2 + bx + c = 0$ is factorable over the real numbers, with integer coefficients in the standard form and factorized form.

I've been thinking, for a quadratic equation to satisfy these properties, its discriminant must be positive and be a perfect square. Let $\displaystyle \Delta_1$ and $\displaystyle \Delta_2$ denote the discriminants of equations one and two. We have :

$\displaystyle \Delta_1 = b^2 - 4(c + n)$

$\displaystyle \Delta_2 = b^2 - 4c$

And I also noted that $\displaystyle \Delta_1 - \Delta_2 = b^2 - 4(c + n) - b^2 + 4c = 4c - 4(c + n) = 4(c - c + n) = 4n$, and this could be useful regarding the difference of two squares.

And so, if I could find the values of $\displaystyle \Delta_1$ and $\displaystyle \Delta_2$, I could easily work out $\displaystyle b$ and $\displaystyle c$ with a simple system of equations. But what I've done beyond this point does not lead to anything interesting, so I'm wondering, do I have enough information about the discriminants to work them out ?

Does anyone have any idea to put me on the right track again ? Thanks all