Hi all,

I have a bit of a unique problem that I'm hoping someone amongst you may have some suggestions on how to obtain a short cut to the answer.

Essentially, I have a list of quadratics that were generated from an initial quadratic equation by using some simple transformation rules.

This family of quadratics (I like to call it a 'family' as they're related to each other because of the transformation rules), will always contain only ONE quadratic equation that has integer roots ... the remaining quadratics in the family have either complex roots or non-integer roots.

Let me use an example to make this clearer ....

lets start with the original quadratic say, $\displaystyle x^2 - 4x + 4894$

Now, lets generate further members of this family by simply adding 24 to the 2nd term and subtracting 1 from the 3rd term ... so now we have the following:

(1) $\displaystyle x^2 - 4x + 4894$ Roots: Complex

(2) $\displaystyle x^2 - 28x + 4893$ Roots: Complex

Essentially, we're using the following: $\displaystyle x^2 - (4x + 24nx) + (4894 - n)$ to generate a family of 16 quadratics using n=0 to 15 iterations.

n

---

(0) $\displaystyle x^2 - 4x + 4894$ Roots: Complex

(1) $\displaystyle x^2 - 28x + 4893$ Roots: Complex

(2) $\displaystyle x^2 - 52x + 4892$ Roots: Complex

(3) $\displaystyle x^2 - 76x + 4891$ Roots: Complex

(4) $\displaystyle x^2 - 100x + 4890$ Roots: Complex

(5) $\displaystyle x^2 - 124x + 4889$ Roots: Complex

(6) $\displaystyle x^2 - 148x + 4888$ Roots: 49.75 and 98.25

(7) $\displaystyle x^2 - 172x + 4887$ Roots: 35.91 and 136.09

(8) $\displaystyle x^2 - 196x + 4886$ Roots: 29.31 and 166.69

(9) $\displaystyle x^2 - 220x + 4885$ Roots: 25.06 and 194.94

(10) $\displaystyle x^2 - 244x + 4884$ Roots: 22 and 222

(11) $\displaystyle x^2 - 268x + 4883$ Roots: 19.66 and 248.34

(12) $\displaystyle x^2 - 292x + 4882$ Roots: 17.80 and 274.20

(13) $\displaystyle x^2 - 316x + 4881$ Roots: 16.29 and 299.71

(14) $\displaystyle x^2 - 340x + 4880$ Roots: 15.02 and 324.98

(15) $\displaystyle x^2 - 364x + 4879$ Roots: 13.94 and 350.06

If we examine the above family of quadratics, we see that the ONLY integer roots of22 and 222occur at iteration10.

Please note that for every starting quadratic that I will be using, that quadraticWILLbe guaranteed to only generate a single subsequent quadratic family member that has integer roots.

Now what I would like, is for someone to suggest a way of QUICKLY finding that quadratic family member with the integer roots. I really don't want to have to plow my way through potentially millions of family members to find that single quadratic that has the only integer roots.

Hopefully I've been reasonably clear in my request and explanation.

Thanks for any assistance ...

stevek