To find the two prime numbers given their sum and product:
The sum of two distinct prime numbers is 85.
x + y = 85
y = 85 - x
Representation:
Let the first prime number be x
The other prime number be 85 - x
The product of the prime numbers is 166.
(x) (85-x) = 166
-x² + 85x = 166
Transform to quadratic equation, ax² +bx = c = 0
-x² + 85x - 166 = 0
Solve by factoring or using quadratic formula. Since the equation can be factored, solve by factoring:
-1(-x² + 85x - 166 = 0)-1
x² - 85x + 166 = 0
(x - 83) (x-2) = 0
x - 83 = 0 x - 2 = 0
x = 83 x = 2
The two prime numbers are 83 and 2.
To check:
83 + 2 = 85
(83) (2) = 166
