What 2 numbers add to 24 and multiply to -180?

Set this up as a system of two equations - that is:
x + y = 24
xy = -180
The easiest way to solve is then via substitution.
y = 24 - x
Which means...
x(24 - x) = -180
Simplify.
24x - x^2 = -180
Solve as a standard quadratic equation. I'll complete the square.
x^2 - 24x = 180
x^2 - 24x + 144 = 180 + 144
(x - 12)^2 = 324
(x - 12) = ±18
x = 30 or -6
This means that your answers are 30 and -6.

x + y = 24
y = 24 - x
x*y = -180
x(24 - x) = -180
24x - x^2 = -180
x^2 - 24x = 180
x^2 - 24x - 180 = 0
(x + 6)(x - 30) = 0
x + 6 = 0 or x - 30 = 0
x = -6 or x = 30
x + y = 24
-6 + y = 24
y = 30
x*y = -180
30y = -180
y = -180/30 = -6

The two numbers are -6 and 30

Lets work on it.
Let x = first number
Let 24 - x = the second number
So
x(24-x) = -180
24x - x^2 = -180
0 = x^2 - 24x - 180
Try quadratic equation:

x = (+24 +- sqr(24^2 - 4*180)) / 2

x = (24 +- sqr(576 - 720))/2
x = (24 +- sqr(-144))/2
Wow, there is no real solution, because there is no
real sqr(-144).
In advanced math,
x = (24 +- 12 i) / 2
x = 12 +-6i
And the other number is the same.

{a+b=24, ab=-180}

From equation 1, we see that a=24-b

a+b=24
a=24-b

Substitute (24-b) for a in equation 2

(24-b)b=-180
-b²-24b=-180
-b²-24b+180=0

-1x²+24x+180=0
{a=-1, b=24, c=180}

x=(-b±√(b²-4ac))/(2a)
=(-(24)±√((24)²-4(-1)(180)))/(2(-1))
=(-24±√(576--720))/-2
=(-24±√1296)/-2
=(-24±36)/-2

x₁=-6
x₂=30

Case 1 (b=-6)

a+b=24
a-6=24
a=30

{a=30, b=-6}

Case 2 (b=30)

a+b=24
a+30=24
a=-6

{a=-6, b=30}

Final Answer,

{{a=30, b=-6}, {a=-6, b=30}}

x + y = 24 or y = 24 - x
xy = -180
substitute first equation into second equation:
x(24 - x) = -180
distribute:
24x - x^2 = -180
-x^2 + 24x + 180 = 0
quadratic formula:
x = [-24 +/- SQR(576 - 4(-1)(180))]/2(-1)
x = [-24 +/- 36]/-2
x = -6, 30
using first equation:
y = 24 - (-6) = 30
y = 24 - 30 = -6
check using second equation:
-6(30) = -180 checks
The numbers are -6 and 30.