Question 188188
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To start with, there is a problem with the way you stated the question.  "...three consecutive negative numbers." doesn't make much sense unless those numbers are integers.  Properly stated, the question should read, "I am thinking of three consecutive negative <i><b>integers</b></i>.  Assuming that is the real case, the solution follows:


Let the first integer be <i>x</i>.  Then the next consecutive integer is <i>x</i> + 1, and the next one is <i>x</i> + 1 + 1 = <i>x</i> + 2.


The first times the second is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x(x + 1) = x^2 + x]


Three times the third is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3(x + 2) = 3x + 6]


Subtract three times the third from the first times the second:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2 + x -(3x + 6) = x^2 + x - 3x - 6 = x^2 - 2x -6]


And this is equal to 57:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2 - 2x - 6 = 57]


Add -57 to both sides:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2 - 2x - 63 = 0]


Factor:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (x + 7)(x - 9) = 0]


So:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x = -7]


or


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x = 9]


Exclude the positive root because we are looking for three consecutive negative integers.  Hence, the first negative integer is -7, the next is -6, and the third is -5:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -7(-6) - 3(-5) = 42 + 15 = 57]


Answer checks.


Note that if the problem had not constrained the solution to negative integers, then 9, 10, and 11 would have worked:  90 - 33 = 57



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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