Harristotelian Logic

I pondered why all those algebraic manipulations just kept giving me equations that relied on my target and realized it was obvious, so I thought about how to get something different, and had a realization.

Let

S = (k_1*sqrt(x) + k_2*sqrt(y))*(k_3*sqrt(x) + k_4*sqrt(y))

where S is the surrogate, and you have to multiply everything out and complete the square, like usual.

But if my idea is a good one, you'll end up with this complicated expression with k_1, k_2, k_3 and k_4 multipled times S, where you would then pick that complicated piece to equal your target composite.

Then you just pick any S, like S=15, and get squares for x and y using it, and then use the sign ambiguity of the square roots.

You see, one way the solution will factor your surrogate S.

The other way, changing the signs, it will factor your target, if this idea works.

Um, can someone solve out that equation for S to get the complicated thing at the end?

It may have special properties that are there just to block this idea!!!

If this idea works, it will be a beautiful demonstration of the inherent sign ambiguity in the square root, as what I thought up is to get an expression that is forced to factor more than one number by use of square roots.