Question 1200102
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Consecutive integers follow one after another.
Examples:
6,7,8
22,23,24


Consecutive even integers are when all of the values are even, and they are as close as possible.
Examples:
2,4,6
100,102,104


Let x be some integer. It might be even or odd.
To guarantee it is always even, we double it to 2x.
Eg: 
x = 5 doubles to 2x = 2*5 = 10
x = 6 doubles to 2x = 2*6 = 12


2x = first even integer
2x+2 = even integer right after 2x
2x+2+2 = 2x+4 = even integer right after 2x+2


In short,
first = 2x
second = 2x+2
third = 2x+4


The sequence {2x,2x+2,2x+4} represents a set of three consecutive even integers, where x is any integer.


We then have the following information "<font color=blue> the product of the first and second is 20 less than 10 times the third</font>".


Let's translate that into an equation and solve for x.


first*second = 10*third - 20
2x*(2x+2) = 10*(2x+4) - 20
4x^2+4x = 20x+40 - 20
4x^2+4x = 20x+20
4x^2+4x-20x-20 = 0 
4x^2-16x-20 = 0
4(x^2-4x-5) = 0
x^2-4x-5 = 0
(x-5)(x+1) = 0
x-5 = 0 or x+1 = 0
x = 5 or x = -1


Optionally you could use the quadratic formula to solve for x.
Let me know if you need to see these steps.


If x = 5, then,<ul><li>2x = 2*5 = 10 is the first even integer</li><li>2x+2 = 2*5+2 = 10+2 = 12 is the second even integer</li><li>2x+4 = 2*5+4 = 10+4 = 14 is the third even integer</li></ul>We end up with the set <font color=red>{10,12,14}</font> as one solution.


If x = -1, then,<ul><li>2x = 2*(-1) = -2 is the first even integer</li><li>2x+2 = 2*(-1)+2 = -2+2 = 0 is the second even integer</li><li>2x+4 = 2*(-1)+4 = -2+4 = 2 is the third even integer</li></ul>We end up with the set <font color=red>{-2,0,2}</font> as the other solution.



Check for the first solution:
first*second = 10*third - 20
10*12 = 10*14 - 20
120 = 140 - 20
120 = 120
The first solution set is confirmed.


Check for the second solution:
first*second = 10*third - 20
-2*0 = 10*2 - 20
0 = 20 - 20
0 = 0
The second solution set is confirmed.
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