How to i manipulate this fraction

[seang]

Homework Statement

I need to make this:

$$\frac{100K}{x^2 + (25 + \alpha)x + (\alpha 25 + 100K)}$$

look like this:

$$\frac{K}{x^2 + \beta x + (K)}$$

Homework Equations

Beta and K can be any combination of variables, as long as the top K and bottom K are equal. A scaling factor is allowed to come outside of the fraction if necessary.

The Attempt at a Solution

 

[HallsofIvy]

The "scaling factor" means that you can multiply either numerator or denominator to make them the same. Multiplying the denominator by something would change the "x²" term and, apparently, you don't want that. That means you must have something of the form
[tex]\frac{1}{C}\frac{100CK}{x^2+ (25+\alpha)x+ (25\alpha+ 100K)}[/itex]<br /> It's pretty obvious what $\beta$ must be! Can you choose C and the <b>new</b> K so that the numerator and constant term in the denominator are the same? (You might want to initially call the new K something like K' to avoid confusion.)[/tex]

 

[seang]

alright, thank you.

But now, for the sake of argument, let's say I wasn't allowed to have a scaling factor outside of the fraction--what could I do then?