# Find two numbers that satisfy these conditions

• Feb 12th 2010, 03:15 AM
Bacterius
Find two numbers that satisfy these conditions
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 :)
• Feb 12th 2010, 07:51 AM
Soroban
Hello, Bacterius!

I've one observation so far . . .

Quote:

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.

$\displaystyle x^2 + bx + c \:=\: 0$ is factorable over the real numbers, with integer coefficients.

$\displaystyle \text{Since }n\text{ is composite, let one factoring be: }\:n \:=\:p\cdot q$

$\displaystyle \text{Let }c = 0\,\text{ and }\,b \:=\:p+q$

$\displaystyle \text{Then both }\:\begin{Bmatrix}x^2 + (p+q)x + pq \\ \\[-3mm] x^2 + (p+q)x\end{Bmatrix}\:\text{ are factorable over the real numbers.}$

• Feb 12th 2010, 01:22 PM
Bacterius
Thank you Soroban, I didn't see this particular case, thanks. Actually, I made a little sign mistake in my original post, it isn't $\displaystyle (c + n)$ but $\displaystyle (c - n)$ in the quadratics and the discriminants, however it does not change your solution. But what if the factorization of $\displaystyle n$ is unknown ?

For example, let $\displaystyle n = 33$. I want to find two numbers $\displaystyle b$ and $\displaystyle c$ that satisfy :
- $\displaystyle b^2 - 4c$ is a perfect square
- $\displaystyle b^2 - 4(c - n)$ is a perfect square

After some guess and check, I find that two numbers that work are $\displaystyle b = 2$ and $\displaystyle c = -15$, because :

$\displaystyle 2^2 - 4 \times (-15) = 64 = 8^2$
$\displaystyle 2^2 - 4 \times (-15 - 33) = 196 = 14^2$

And thus, $\displaystyle x^2 + 2x - 15$ is factorable over the reals with integer coefficients, as well as $\displaystyle x^2 + 2x - 48$.

But there is a problem : I used guess and check. That's why I was looking for a particular process to find $\displaystyle b$ and $\displaystyle c$, more efficient than trial.
• Feb 13th 2010, 09:17 PM
Bacterius
Ok, so this is the actual problem, reformulated in a easier to understand yet equivalent way :

I am looking for two integers, $\displaystyle x$ and $\displaystyle y$.
I know the value of some integer $\displaystyle n$, that can vary.
I also know that $\displaystyle x^2 = b^2 - 4(c - n)$ and $\displaystyle y^2 = b^2 - 4c$ for some integers $\displaystyle b$, $\displaystyle c$.
I also know that $\displaystyle x^2 - y^2 = b^2 - 4(c - n) - (b^2 - 4c) = b^2 - 4c + 4n - b^2 + 4c = 4n$.

Do I have enough information to find $\displaystyle x$ and $\displaystyle y$ ? Or do I need some other information like their sum or product ?
• Feb 14th 2010, 01:54 AM
SuperCalculus
Argh asdf this is annoying. It seems to be that you need to use some form of substitution, but everything I try seems to go around in circles.
• Feb 14th 2010, 02:04 AM
Bacterius
Quote:

Originally Posted by SuperCalculus
Argh asdf this is annoying. It seems to be that you need to use some form of substitution, but everything I try seems to go around in circles.

I know how you feel, having fiddled with algebra for ten minutes just to find that $\displaystyle 0 = 0$.
• Feb 14th 2010, 02:10 AM
SuperCalculus
Quote:

Originally Posted by Bacterius
I know how you feel, having fiddled with algebra for ten minutes just to find that $\displaystyle 0 = 0$.

Or to find that $\displaystyle x^2 = y^2 + 4n$for the fifteenth time.