b is the other number
a + b = -35
a * b = -3800
from the first equation, solve for b in terms of a to get b = -a - 35
in the second equation, replace b with -a - 35 to get:
a * (-a - 35) = -3800
remove parentheses to get:
-a^2 - 35a = -3800
add 3800 to both sides of this equation to get:
-a^2 - 35a + 3800 = 0
multiply both sides of this equation by -1 to get:
a^2 + 35a - 3800 = 0
this is a quadratic equation that can be factored to get:
a1 = 46.58002809
a2 = -81.58002809
you know that:
a1 + b1 = -35
a2 + b2 = -35
solve for b1 and b2 to get:
b1 = -81.58002809
b2 = 46.58002809
you get:
a1 = 46.58002809 and b1 = -81.58002809
a2 = -81.58002809 and b2 = 46.58002809
what this says is:
if a = 46.58..., then b = -81.58...
if a = -81.58..., then b = 46.58...
these equaations satisfy the two equations we started with simultaneously.
those equations are:
a + b = -35
a * b = -3800
we can graph these equations and we can then see the solution graphically.
let x = a and y = b,
you get x + y = -35
x * y = -3800
solve for y in both equations to get:
y = -35 - x
y = -3800 / x
the intersections of the graph of these equations is our solution because that point is common to both equations.
the graph look like this:
