Question

Consider Cardano's problem of finding two numbers whose sum is 10 and whose product is 40.

a) Cardano knew beforehand that no such (real) numbers existed. How did he know? Can you prove it?

b) Solve the system of equations x+y=10 and xy=40 to find Cardano's complex solution.

c) Check that this solution does work-that is, thatb the sum of your complex numbers is 10 and that their product is 40.

Answer 1

a)These two numbers must be both positive. If they are both negative, their product could be 40 but then their sum would be a negative number, not 10. If they are one positive and one negative, then their product would be a negative number, not 40.

Two positive integers whose sum is 10. Their product would be the greatest when they are equal. This is a fact my son learnt from geometry: of all the rectangles with same perimeter, square is the one with greatest area.

So in this case, the biggest product of these two numbers would be 5 x 5 = 25. Since 40 is higher than 25, this is impossible.

b)So first I'd solve for y in one equation.

x+y=10

y= 10-x

Then I'd put that in the other equation.
xy=40

x(10-x) = 40

-x^2 + 10x = 40

x^2 - 10x +40=0

Now do the quadratic formula.

The discriminant is negative.

100-4(1)(40)= -60

Which have two imaginary roots :
x = 5 + sqrt(15)i
x = 5 - sqrt(15)i