The sum of the squares of two consecutive odd integers is 1570. find the integers

Let's look at a few examples of two consecutive odd integers.

1, 3

11, 13

55, 57

Notice that the larger integer is always 2 more than the smaller one.

If you let the smaller one be x, then the larger one is x + 2.

Now we square the two numbers.

The square of x is x^2.

The square of x + 2 is (x + 2) ^2.

Now we add those two squares and set equal to 1570.

x^2 + (x + 2) ^2 = 1570

x^2 + x^2 + 4x + 4 = 1570

2x^2 + 4x + 4 = 1570

2x^2 + 4x - 1566 = 0

x^2 + 2x - 783 = 0

783 is 290 * 27

(x - 27) (x + 29) = 0

x - 27 = 0 or x + 29 = 0

x = 27 or x = - 29

Remember that x is the smaller of the two integers.

If x = 27, then x + 2 is 29.

If x = - 29, then x + 2 is - 27.

The answer is:

There are two pairs of integers

One pair is 27 and 29.

The other pair is - 29 and - 27.