What two numbers add to get 10 and multiply to get - 60

The numbers are "x" and "y"

Therefore, we can suggest this system of equations:

x+y=10 ⇒x=10-y

xy=-60

Solve this system of equations by substitution method.

(10-y) y=-60

10y-y²=-60

y²-10y-60=0

Solve this square equation:

y=[10⁺₋√ (100+240) ]/2 = (10⁺₋2√85) / 2=5⁺₋√85

We have two solutions:

y₁=5-√85 ⇒x₁=10 - (5-√85) = 5+√85

y₂=5+√85 ⇒x₂=10 - (5+√85) = 5-√85

Answer: the numbers are (5-√85) and (5+√85)