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

the numbers are (5-√85) and (5+√85)

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