Question 1199644
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Answer: <font size=4 color=red>14, 16, 18</font>


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Explanation:


Consecutive integers follow one after another. 
Example: 4, 5, 6
Each adjacent neighboring item has a gap of 1.


Consecutive even integers are the same idea, but all of the values must be even. 
Example: 8, 10, 12
Each adjacent neighboring item has a gap of 2.


Let x be a positive even number from the set {2, 4, 6, 8, ...}


x = 1st even integer
x+2 = 2nd even integer = median = middle
x+4 = 3rd even integer
The gap from x to x+2 is +2, and the gap from x+2 to x+4 is also +2


(x+2)*(x+4) = "product of median and the largest integer"
21x-6 = "Six less than 21 times the smallest integer" 


"The product of the median and largest integer is 6 less than 21 times the smallest integer" translates to
(x+2)*(x+4) = 21x-6


We'll expand things out and get everything to one side
(x+2)*(x+4) = 21x-6
x^2+6x+8 = 21x-6
x^2+6x+8-21x+6 = 0
x^2-15x+14 = 0


Then you have a few options at this point.


One method is to factor like so:
x^2-15x+14 = 0
(x-1)(x-14) = 0
x-1 = 0 or x-14 = 0
x = 1 or x = 14


We ignore x = 1 since x must be even.
Therefore, x = 14 is the only solution.


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Another method of solving:


We can apply the quadratic formula
x^2-15x+14 = 0 is of the form ax^2+bx+c = 0
where,
a = 1
b = -15
c = 14
So,
{{{x = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-(-15)+-sqrt((-15)^2-4(1)(14)))/(2(1))}}}


{{{x = (15+-sqrt(225-56))/(2)}}}


{{{x = (15+-sqrt(169))/(2)}}}


{{{x = (15+-  13)/(2)}}}


{{{x = (15+13)/(2)}}} or {{{x = (15-13)/(2)}}}


{{{x = (28)/(2)}}} or  {{{x = (2)/(2)}}}


{{{x = 14}}} or  {{{x = 1}}}
We arrive at the same two solutions found earlier. 
And as mentioned earlier, we ignore x = 1 to go for x = 14.


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A third method:


Graph out y = x^2-15x+14 using a TI calculator, Desmos, or GeoGebra. 
There are tons of options out there so feel free to use your favorite graphing calculator.


{{{graph(500,500,-3,16,-50,50,0,x^2-15x+14)}}}


The parabola crosses the x axis at the locations (1,0) and (14,0) which shows that x = 1 and x = 14 are the two solutions or roots.


The term "x intercept" is another way of saying "root" or "zero of a function". 


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If x = 14, then,
x+2 = 14+2 = 16
x+4 = 14+4 = 18


So that's how we arrive at <font size=4 color=red>14, 16, 18</font> as the final answer.


Check:
median*largest = 16*18 = 288
21*smallest-6 = 21*14-6 = 288
This confirms that the equation (x+2)(x+4) = 21x-6 is correct when x = 14, and confirms the final answer.
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