Question 188235
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Your question doesn't make much sense unless you specify that the numbers are integers, whole numbers, natural numbers, or counting numbers, because 'consecutive' doesn't have any meaning unless you are dealing with these sorts of numbers.


Presuming that you meant to say:  "The product of two consecutive <i><b>integers</b></i> is 14 less than 10 times the smaller integer. Find each integer," proceed thus:


Let the first integer be <i>x</i>.  Then the next consecutive integer must be <i>x</i> + 1.


The product of the two integers is then:


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


Ten times the smaller integer is 10<i>x</i>.  And 14 less than that is 10<i>x</i> - 14, so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x^2 + x = 10x - 14] 


Put the quadratic into standard form:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x^2 - 9x + 14 = 0]


Since -7 + (-2) = -9 and (-7)(-2) = 14, this factors:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  (x - 2)(x - 7) = 0] 


Hence,


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x - 2 = 0 \ \ \Rightarrow\ \ x = 2] 


or


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x - 7 = 0 \ \ \Rightarrow\ \ x = 7] 


So your numbers are either 2 and 3 or 7 and 8.


Check:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2 \times 3 =^? 10(2) - 14]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 7 \times 8 =^? 10(7) - 14]


You can do the arithmetic to check both sets of answers.


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