Question

Numbers that add to -0.5 and multiply to 1

Answer 1

Short Answer
There are two numbers
x1 = -0.25 + 0.9682i   <<<< answer 1
x2 = - 0.25 - 0.9582i   <<<< answer 2 
 
I take it there are two such numbers.
Let one number = x
Let one number = y

x + y = -0.5    
y = - 0.5 - x       (1)
xy = 1               (2)

Put equation 1 into equation 2
xy = 1
x(-0.5 - x) = 1
-0.5x - x^2 = 1               Subtract 1 from both sides.
-0.5x - x^2 - 1 = 0          Order these by powers
-x^2 - 0.5x -1 = 0           Multiply though by - 1
x^2 + 0.5x + 1 = 0         Use the quadratic formula to solve this.

$\text{x = }\dfrac{ -b \pm \sqrt{b^{2} - 4ac } }{2a}$

a = 1
b = 0.5
c = 1


$\text{x = }\dfrac{ -0.5 \pm \sqrt{0.5^{2} - 4*1*1 } }{2*1}$

x = [-0.5 +/- sqrt(0.25 - 4)] / 2
x = [-0.5 +/- sqrt(-3.75)] / 2
x = [-0.25 +/- 0.9682i

x1 = -0.25 + 0.9682 i
x2 = -0.25 - 0.9682 i

These two are conjugates. They will add  as x1 + x2 = -0.25 - 0.25 = - 0.50.

The complex parts cancel out.  Getting them to multiply to 1 will be a little more difficult. I'll do that under the check.

Check
(-0.25 - 0.9682i)(-0.25 + 0.9682i)
Use FOIL
F:-0.25 * -0.25 = 0.0625
O: -0.25*0.9682i
I: +0.25*0.9682i
L: -0.9682i*0.9682i = - 0.9375 i^2 = 0.9375

Notice
The two middle terms (labled "O" and "I" ) cancel out. They are of opposite signs.

The final result is 0.9375 and 0.0625 add up to 1