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

Question

Eli Assouline · Accepted Answer

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

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 ext{ Expand left hand side}# #x^2 -13x + 22 = 0 ext{ Rearrange into quadratic form}# #(x-11)(x-2) = 0 ext{ 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#

Isaiah Anthony · Answer

undefined The two consecutive odd integers are 7 and 9.

Isaiah Anthony · Answer

undefined Let $ x $ represent the smaller odd integer. Then the next consecutive odd integer would be $ x + 2 $. According to the given conditions:

$$ x(x + 2) = 15x - 22 $$

Expanding and rearranging terms:

$$ x^2 + 2x = 15x - 22 $$
$$ x^2 - 13x + 22 = 0 $$

Factoring or using the quadratic formula:

$$ (x - 11)(x - 2) = 0 $$

The solutions for $ x $ are $ x = 11 $ or $ x = 2 $. However, since we're considering odd integers, the correct solution is $ x = 11 $. Therefore, the smaller integer is 11 and the next consecutive odd integer is $ 11 + 2 = 13 $. So, the integers are 11 and 13.