What two numbers add to ten and multiply to forty?
I think the answer includes radicals and/or imaginary numbers.
If a + b = 10 then b = 10 - a. Thus ab = a(10 - a) = 10a - a 2. If you plot the graph of
f(a) = 10a - a 2
you will see that it is a parabola that opens downward with its vertex at (5,25). Hence if a and b are real numbers with a + b = 10 then the largest value ab can attain is 25.
Suppose a and b are complex numbers, a = x + iy and b = s + it, then
a + b = (x + s) + i(y + t)
Since a + b is real, t = -y. Thus
a = x + iy, b = s - iy and a + b = x + s.
Now calculate ab. What do you know about s if ab is real? Now can you solve your original problem?
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