Answer:
10/(1+√11), 1+√11 and 10/(1-√11) and 1-√11
Explanation:
Let x be the first number and y the second number
two numbers that multiplied together give me 10
x*y = 10
and the difference of the two numbers is -2?
x-y = -2
we can write x in terms of y so we only have one variable y
x-y =-2 , add y to both sides
x-y+y = y -2, combine like terms
x= y-2
x*y = 10, substitute x=y-2
(y-2)y = 10, distribute in parenthesis
y² -2y-10 = 0, subtract 10 from both sides
We can use the quadratic formula to solve y² -2y-10 = 0
y = (-b±√b²-4ac)/2a, where a= 1, b= -2, c= -10
y = (2±√4+4*1*10)/2
y = (2±√44)/2
y= (2±2√11)/2
y= 1±√11
x*y = 10, divide both sides by y
x= 10/y
x= 10/(1±√11)
Check :
x*y = 10
[10/(1±√11)] * (1±√11) = 10
⇒[10/(1+√11)] * (1+√11) = 10✅
⇒[10/(1-√11)] * (1-√11) = 10✅
⇒[10/(1+√11)] * (1-√11) = 10❌
⇒[10/(1-√11)] * (1+√11) = 10❌
x-y = -2
⇒
[10/(1+√11) ] -(1+√11) =
[10 -(1+√11 )(1+√11 )]/ (1+√11 ) =
[10 - (1+11+2√11)]/ (1+√11) =
(10-12 -2√11) / (1+√11) =
(-2 -2√11) / (1+√11) =
-2(1+√11)/ (1+√11) =
-2 ✅
⇒
[10/(1-√11) ] -(1-√11) =
[10 -(1-√11 )(1-√11 )]/ (1-√11 ) =
[10 - (1+11-2√11)]/ (1-√11) =
(10-12 +2√11) / (1-√11) =
(-2 +2√11) / (1-√11) =
-2(1 -√11)/ (1-√11) =
-2 ✅