Question

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

Answer 1

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:
`(x)(x+2) = 15x - 22`

Solve for `x` to find the smaller integer
`x^2 + 2x = 15x - 22 \text{ Expand left hand side}`
`x^2 -13x + 22 = 0 \text{ Rearrange into quadratic form}`
`(x-11)(x-2) = 0 \text{ Solve quadratic equation}`

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`