The product of 2 positive, consecutive odd numbers is 255 .

Question

Part A: Write a quadratic equation in standard form to represent this problem

Part B: What are the 2 numbers?

Part C: Show your work for Part B.

Answer 1

(2k + 1)(2k + 3) = 255, assuming that k > 0.

4k^2 + 8k + 3 = 255

4k^2 + 8k - 252 = 0

k^2 + 2k - 63 = 0

Solve for k, by factoring

(k + 9)(k - 7) = 0

So, either

k + 9 = 0

k = -9

or

k - 7 = 0

k = 7.

Since we assumed that k > 0, we can eliminate the negative

k = 7, so the two consecutive positive odd numbers are 15 and 17.

Answer 2

To guarantee an even number we can use the expression "2n" where n is any integer. So, to guarantee an odd number we can use the expression "2n + 1".

That means that the next consecutive odd integer will be "2n + 3"

The product of the odd integers is (2n + 1)(2n + 3) and that equals 255:

(2n + 1)(2n + 3) = 255

4n² + 8n + 3 = 255

4n² + 8n - 252 = 0

n² + 2n - 63 = 0

(n + 9)(n -7) = 0

n = -9 and n = 7

We are told that the numbers must be positive so we can throw out n = -9

If n = 7, then the first odd number = 2(7) + 1 = 15 and the second is then 17