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The sum of two positive numbers is 5 and the sum of their cubes is 35. What is the sum of their squares?

Sagot :

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