The product of two consecutive even integers is 624. How do you find the integers?

Question

Genesis Allen · Accepted Answer

See a solution process below:

First, let's call the first number: #x# Then the next consecutive even integer would be: #x + 2# Therefore their product in standard form would be: #x(x + 2) = 624# #x^2 + 2x = 624# #x^2 + 2x - color(red)(624) = 624 - color(red)(624)# #x^2 + 2x - 624 = 0# We can factor this as: (x + 26)(x - 24) = 0 Now, we can solve each term on the left side of the equation for #0#: Solution 1: #x + 26 = 0# #x + 26 - color(red)(26) = 0 - color(red)(26)# #x + 0 = -26# #x = -26# Solution 2: #x - 24 = 0# #x - 24 + color(red)(24) = 0 + color(red)(24)# #x - 0 = 24# #x = 24# If the first number is #-26# then the second number is: #-26 + 2 = -24# #-26 * -24 = 624# If the first number is 24 then the second number is: #24 + 2 = 26# #24 * 26 = 624# There are two solutions to this problem: #{-26, -24}#; #{24, 26}#

Genesis Allen · Answer

undefined Let $ n $ represent the smaller of the two consecutive even integers. Then the next consecutive even integer is $ n + 2 $.

Since the product of the two consecutive even integers is 624, we can set up the equation:

$$ n \cdot (n + 2) = 624 $$

Expanding the left side of the equation:

$$ n^2 + 2n = 624 $$

Rearranging the equation into standard quadratic form:

$$ n^2 + 2n - 624 = 0 $$

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.

For simplicity, let's use factoring:

$$ (n + 26)(n - 24) = 0 $$

This gives us two possible solutions:

1. $ n + 26 = 0 $  -->  $ n = -26 $ (not valid for consecutive even integers)
2. $ n - 24 = 0 $  -->  $ n = 24 $

So, the smaller of the two consecutive even integers is 24, and the next consecutive even integer is $ 24 + 2 = 26 $. Therefore, the two consecutive even integers are 24 and 26.