I have this problem:
Find three odd consecutive numbers with the property that the product of the first one and the third one minus the product of the first one and the second is greater by eleven than the third one.
I have solved the problem with the following equation:
$(2x+1)(2x+5)-(2x+1)(2x+3)=11 + (2x+5).$
Solution is $x=7$ which gives $15,17,19$ as the requested numbers.
Now If use simpy $x, x+2, x+4$ to denote these numbers I also obtain the same solution with the equation:
$x(x+4)-x(x+2)=11+(x+4)$.
Solution is $x=15$ so the numbers are $15, 17, 19$.
So my doubt is why I didnt have the need to express the numbers as a proper odd number $(2x+1)$ and it works simply with $x$.
Could be the case that there exist some other three (even) numbers that satisfy the conditions and so it's ok to use the proper expression for an odd number. Or can I always use $x, x+2, x+4, x+6$ to denote consecutive odd numbers without getting into any problems.
My question is why I was able to find the same solution using $x, x+2, x+4$ to denote the numbers, why I didn't have to use the proper $2x+1$. This tells me I can always use $x, x+2, x+4..$ to denote consecutive odd numbers to solve this kind of problems. Probably it's just a coincidence.
Answer
Your first equation requires that the numbers be odd if $x$ is a whole number. Your second does not. If you find a whole solution to the first, you are done. In theory, you could find a whole solution to the second in which $x$ was even. You would have to check the solution(s) you find to make sure the numbers were odd. Alternatively, you could look at your second equation and note that if $x$ is even all the terms are even except for the $11$, so the equation will fail. This guarantees that all solutions will have $x$ odd, satisfying the constraint. There is nothing wrong with the second approach if you check the solution(s) you find to be odd, or prove that all solutions are odd. It will find all the solutions, and maybe more.
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