x + y = 6
x*y = 12
It's easy to solve for one variable, substitute into the other equation, and use the quadratic formula. In this case, we get a negative radical and two complex numbers as solutions.
We could also graph and see that the line y = 6 - x does not intersect the hyperbola y = 12/x. That only tells us there is no real solution. It does show us that in such a system, if the sum of two numbers is less than their product, the numbers must be complex.
Loh's method says we should assume the solution is centered around the sum.
So x = 3 + d and y = 3 - d.
When we substitute into the other equation, we get a simpler quadratic, 9 - d^2 = 12, which simplifies to
d = +/- sqrt(-3) = +/- i*sqrt(3)
So are answers are x = 3 +/- i*sqrt(3) and y = 3 -/+ i*sqrt(3)
Yes, it is possible to have a reverse plus/minus sign. Here it shows how the answers properly pair.
It's good to get the right answer and better to get it quickly. See Karatsuba's multiplication algorithm for a more complex example.
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