Question

What two numbers multiply to 10 and add to -29 ?

Answer 1

There are none.
You can prove this by graphing.
xy = 10
y = 10/x

x + y = -29
y = -x - 29

On the graphing calculator, type in
Y1 = 10/x
Y2 = -x - 29

As you can see the two do not intersect.

Answer 2

xy=10
x+y=29

subtract x from both sides for second equation

y=-x-29
subsitute in second equaiton
x(-x-29)=10
-x^2-29x=10
add x^2+29x to both sides
x^2+29x+10=0
quadratic formula
if you have
ax^2+bx+c=0,
x=
`\frac{-b+/- \sqrt{b^(2)-4ac} }{2a}`

so
x^2+29x+10=0
a=1
b=29
c=10
x=
`\frac{-29+/- \sqrt{(-29)^(2)-4(1)(10)} }{2(1)}`
x=
`(-29+/- √(841-40) )/(2)`
x=
`(-29+/- √(801) )/(2)`
x=
`(-29+/- 3√(89) )/(2)`

x=
`(-29- 3√(89) )/(2)` or
`(-29+ 3√(89) )/(2)`
those are the 2 numbers
aprox=-28.65 and -0.349