# If the product of two consecutive odd integers is decreased by 3, the result is 12. How do you find the integers?

the integers are $3$ and $5$
Also the integers are $- 3$ and $- 5$

#### Explanation:

The solution

Let $2 n + 1$ be the odd integer
Let $2 n + 3$ be the next odd integer

$\left(2 n + 1\right) \left(2 n + 3\right) - 3 = 12$

$4 {n}^{2} + 8 n + 3 - 3 = 12$

$4 {n}^{2} + 8 n = 12$

${n}^{2} + 2 n = 3$

${n}^{2} + 2 n - 3 = 0$

Solution by factoring

$\left(n - 1\right) \left(n + 3\right) = 0$

$n - 1 = 0$
$n = 1$

Let $2 n + 1$ be the odd integer which is equal $= 3$
Let $2 n + 3$ be the next odd integer which is equal $= 5$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

At $n + 3 = 0$

$n = - 3$
Let $2 n + 1$ be the odd integer which is equal $= - 5$
Let $2 n + 3$ be the next odd integer which is equal $= - 3$

Checking: using odd numbers 3, and 5
$\left(2 n + 1\right) \left(2 n + 3\right) - 3 = 12$
$\left(3\right) \left(5\right) - 3 = 12$
$12 = 12 \text{ }$correct

Checking: using odd numbers -5, and -3
$\left(2 n + 1\right) \left(2 n + 3\right) - 3 = 12$
$\left(- 5\right) \left(- 3\right) - 3 = 12$
$12 = 12 \text{ }$correct

God bless....I hope the explanation is useful.