SOLUTION: The product of two consecutive even integers is 288. find the integers

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Question 100405: The product of two consecutive even integers is 288. find the integers
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Consecutive even integers follow the form: 2x, 2x%2B2, 2x%2B4, ...., etc.


So the product of two consecutive even integers is:


2x%282x%2B2%29=288


4x%5E2%2B4x=288 Distribute


4x%5E2%2B4x-288=0 Subtract 288 from both sides


Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve 4%2Ax%5E2%2B4%2Ax-288=0 ( notice a=4, b=4, and c=-288)




x+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A4%2A-288+%29%29%2F%282%2A4%29 Plug in a=4, b=4, and c=-288



x+=+%28-4+%2B-+sqrt%28+16-4%2A4%2A-288+%29%29%2F%282%2A4%29 Square 4 to get 16



x+=+%28-4+%2B-+sqrt%28+16%2B4608+%29%29%2F%282%2A4%29 Multiply -4%2A-288%2A4 to get 4608



x+=+%28-4+%2B-+sqrt%28+4624+%29%29%2F%282%2A4%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-4+%2B-+68%29%2F%282%2A4%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%28-4+%2B-+68%29%2F8 Multiply 2 and 4 to get 8

So now the expression breaks down into two parts

x+=+%28-4+%2B+68%29%2F8 or x+=+%28-4+-+68%29%2F8

Lets look at the first part:

x=%28-4+%2B+68%29%2F8

x=64%2F8 Add the terms in the numerator
x=8 Divide

So one answer is
x=8



Now lets look at the second part:

x=%28-4+-+68%29%2F8

x=-72%2F8 Subtract the terms in the numerator
x=-9 Divide

So another answer is
x=-9

So our solutions are:
x=8 or x=-9



Now plug in x=8 to find the two numbers

2%288%29=16
2%288%29%2B2=18

So one pair of numbers is 16 and 18



Now plug in x=-9 to find the two numbers

2%28-9%29=-18
2%28-9%29%2B2=-16

So another pair of numbers is -18 and -16