The product of two consecutive even integers is 36 less than 18 times their sum. Find the two integers

Answers:

There are two possible outcomes

If your teacher wants only positive integers, then go for the second scenario.

Explanation:

x = smaller integer

x+2 = larger integer

We can see that x+2 follows directly after x, so they are consecutive. Also, there's a gap of 2 units between the values. This is due to the phrasing "consecutive even" instead of just "consecutive".

Their product is x(x+2) = x^2+2x

Their sum is x+(x+2) = 2x+2

The product is 36 less than 18 times their sum, which means,

product = 18*(sum) - 36

x^2+2x = 18(2x+2) - 36

x^2+2x = 36x+36-36

x^2+2x = 36x

x^2+2x-36x = 0

x^2-34x = 0

x(x-34) = 0

x = 0 or x-34 = 0

x = 0 or x = 34

If x = 0, then x+2 = 0+2 = 2

The product is x*(x+2) = 0*2 = 0

The sum is x+(x+2) = 0+2 = 2

If we multiply the sum by 18, then we get 18*2 = 36. Subtracting off 36 leads to 36-36 = 0

Therefore the equation below

product = 18*(sum) - 36

has been satisfied

If x = 34, then x+2 = 34+2 = 36

product = x*(x+2) = 34*36 = 1224

sum = x+(x+2) = 34+36 = 70

then we can see that...

product = 18*(sum) - 36

1224 = 18*(70) - 36

1224 = 1260 - 36

1224 = 1224

So that works as well.

There are two possible outcomes

Either the integers are 0 and 2

OR

the integers are 34 and 36