The two numbers: x and y
x + y = 5 ⇒ y = 5 - x
Representation:
x = first number
5 - x = second number
Sum of their cubes:
(x)³ + (5-x)³ = 35
x³ + 125 - 75x + 15x² - x³ = 35
x³ - x² + 15x² - 75x + 125 = 35
15x² - 75x + 125 = 35
Transform to Quadratic Equation form, ax² + bx + c = 0
15x² - 75x + 125 - 35 = 0
15x² - 75x + 90 = 0
Factor out the GCF of each term: 15
15 ( x² - 5x + 6) = 0
Factor x² - 5x + 6:
(x - 2) (x - 3) = 0
x - 2 = 0
x = 2
x - 3 = 0
x = 3
The two positive numbers are 2 and 3
The sum of their squares:
Sum of their squares = (2)² + (3)²
Sum of their squares = 4 + 9
Sum of their squares = 13
The sum of the squares of the 2 and 3 is 13.
To check:
Sum of the two positive numbers 2 and 3 is 5
2 + 3 = 5
Sum of the cubes of the two positive numbers 2 and 3 is 35.
(2)³ + (3)³ = 35
(2)(2)(2) + (3)(3)(3) = 35
8 + 27 = 35
35 = 35