What are the numbers that have 12 as sum and 40 as product

Question

What are the numbers that we can choose to have 12 as sum and both 40 as product?

Answer 1

X+y = 12

xy = 40

x(12 - x) = 40

12x - x^2 = 40

0 = x^2 - 12x + 40

x = [ 12 +or - sqrt ( 144 - 4*40) ]/2

= [ 12 +or- sqrt( 144 - 160)]/2

= [ 12 +or- sqrt(-16) ]/ 2

= [ 12 +or- 4i]/2

= 6 +or- 2i

12 - (6 + 2i) = 12 - 6 - 2i = 6 - 2i

So (6 + 2i) and (6 - 2i)

CHECK:

the sum is clearly 12.

(6 + 2i) ( 6 - 2i) = 36 - (4)(-1) = 36 + 4 = 40

It turns out that if the first number chosen is 6-2i then

the other number is also the conjugate, as 12 - (6 - 2i) = 12 - 6 + 2i = 6 + 2i

6 + 2i and 6 - 2i