The product of two consecutive odd integers is 22 less than 15 times the smaller integer. What are the integers?

Jun 20, 2016

The two integers are $11$ and $13$.

Explanation:

If $x$ represents the smaller integer, the larger integer is $x + 2$, as the integers are consecutive and $2 +$ an odd integer will give the next odd integer.

Converting the relationship described in words in the question into a mathematical form gives:
$\left(x\right) \left(x + 2\right) = 15 x - 22$

Solve for $x$ to find the smaller integer
${x}^{2} + 2 x = 15 x - 22 \setminus \textrm{E x p \mathmr{and} \le f t h \mathmr{and} s i \mathrm{de}}$
${x}^{2} - 13 x + 22 = 0 \setminus \textrm{R e a r r a n \ge \int o \quad r a t i c f \mathmr{and} m}$
$\left(x - 11\right) \left(x - 2\right) = 0 \setminus \textrm{S o l v e \quad r a t i c e q u a t i o n}$

The quadratic equation is solved for
$x = 11$ or $x = 2$

As the question specifies the integers be odd, $x = 11$ is the only useful solution.

The smaller integer is $x = 11$
The larger integer is $x + 2 = 13$