find consecutive odd positive integers

by bibi
(city)

The sum of squares of two consecutive odd positive integers is 202. Find the integers.

Solution

Let 2x + 1 be the first odd integer

Then, 2x + 3 will be the second integer

Now square each odd integer

(2x + 1)² = (2x + 1) × (2x + 1)


(2x + 1) × (2x + 1) = 2x × 2x + 2x × 1 + 1 × 2x + 1 × 1


(2x + 1) × (2x + 1) = 4x² + 2x + 2x + 1 = 4x² + 4x + 1


(2x + 3)² = (2x + 3) × (2x + 3)


(2x + 3) × (2x + 3) = 2x × 2x + 2x × 3 + 3 × 2x + 3 × 3


(2x + 3) × (2x + 3) = 4x² + 6x + 6x + 9 = 4x² + 12x + 9


Since the sum of the squares equal 202, we get

4x² + 4x + 1 + 4x² + 12x + 9 = 202


8x² + 16x + 10 = 202

8x² + 16x + 10 - 202 = 202 - 202


8x² + 16x - 192 = 0


Divide everything by 8

We get x² + 2x - 24 = 0

To factor the left side of this equation, look for factors of -24 that will add up to 2.

6 × -4 = -24 and 6 + -4 = 2.

We get (x + 6) × (x + -4) = 0


We have two equation to solve

x + 6 = 0 and x + -4 = 0

x = -6 and x = 4

x = -6 cannot be used since the integers are positive

we can use x = 4.

The first integer is 2x + 1 = 2 × 4 + 1 = 9

The seond integer is 2x + 3 = 2 × 4 + 3 = 11

Indeed 9² + 11² = 81 + 121 = 202