how to find integers

Question

The product of two consective odd integers is 77 more than twice the larger,find the integers.

Answer 1

Let the two integers be A and B.

Product: AB = 77 + 2A, where A is the larger of the two integers.

Since A and B are consecutive odd integers, then A = B + 2.

Substituting for A = B + 2 into the Product equation,

(B + 2)B = 77 + 2(B + 2)

B^2 + 2B = 81 + 2B

B^2 = 81

B = +/- 9

If B = 9, then A = 11, AB = 99, which is 77 greater than 2A (=22), the larger of the two integers. (A valid solution)

If B = -9, then A = -7, AB = 63, which is 77 greater then 2A (= -14), the larger of the two integers. (A valid aolution)

There are thus two valid solutions.

(A, B) = (11, 9) and (A, B) = (-7, -11)