Question

1) The sum of the squares of two consequtive odd positive integers is 290. Find the numbers.

2) The product of two consequtive even natural numbers is 224. Find the number.

Answer 1

1) Answer
Let one of the odd positive integer be x
then the other odd positive integer is x+2
their sum of squares = x² +(x+2)²
= x² + x² + 4x +4
= 2x² + 4x + 4
Given that their sum of squares = 290
⇒ 2x² +4x + 4 = 290
⇒ 2x² +4x = 290-4 = 286
⇒ 2x² + 4x -286 = 0
⇒ 2(x² + 2x - 143) = 0
⇒ x² + 2x - 143 = 0
⇒ x² + 13x - 11x -143 = 0
⇒ x(x+13) - 11(x+13) = 0
⇒ (x-11) = 0 , (x+13) = 0
Therfore , x = 11 or -13
We always take positive value of x
So , x = 11 and (x+2) = 11 + 2 = 13
Therefore , the odd positive integers are 11 and 13 .

2) Answer

Let one integer be x, and the other be (x+2)
x(x+2)=224
x²+2x-224=0
(x+16)(x-14) =0
x=-16 or 14

so x+2 = -14 or 16
So the integers are either 14 and 16 or -14 and -16.