Question

the sum of the squares of three consecutive positive integers is 50.find the integers

Answer 1

Let the three consecutive positive  no`s be  x, x + 1, x + 2
then the sum of the squares of the no`s = x² + (x + 1)² + (x + 2)² = 50 as given
x² + x² + 1 + 2x + x² + 4 + 4x = 50
⇒3x² + 6x + 5 = 50
⇒3x² + 6x = 45
⇒3x² + 6x - 45 = 0
⇒3x² -9x + 15x - 45 = 0
⇒3x(x - 3) + 15(x - 3) = 0
⇒(x - 3)(3x + 15) = 0
⇒x - 3 = 0  or   3x + 15 = 0
⇒x = 3   or   3x = -15
⇒x = 3   or   x = -15/3 = -5
as the question describes the no`s as positive x =3 and not -5
∴x = 3  , x + 1 =4 , x+2 = 5
 ∴the no`s are 3,4,5. 

Answer 2

Let one of the numbers = x
and other two numbers are = x-1 and x+1

(When you have to take three consecutive numbers, always choose x-1, x and x+1. calculations will become easy;)

sum of squares = 50
⇒ (x-1)² + x² + (x+1)² = 50
⇒ (x²-2x+1) + x² + (x²+2x+1) = 50
⇒ x²-2x+1 + x² + x²+2x+1 = 50
⇒ 3x² + 2 = 50
⇒ 3x² = 50-2 = 48
⇒ x² = 48/3 = 16
⇒ x = √16 
⇒ x = 4
x-1 = 4-1=3
x+1=4+1=5
three numbers are 3,4,5