SOLUTION: Find two consecutive positive integers such that the square of the first increased by 2 times the second is equal to 37.

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Question 1208747: Find two consecutive positive integers such that the square of the first
increased by 2 times the second is equal to 37.

Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find two consecutive positive integers such that the square of the first
increased by 2 times the second is equal to 37.
~~~~~~~~~~~~~~~~~~~~~ 

Let first (smaller) positive integer be n; then the next consecutive integer number is (n+1).


Write an equation according to the problem

    n^2 + 2*(n+1) = 37.


Simplify

    n^2 + 2n + 2 = 37,

    n^2 + 2n - 35 = 0


Factorize

    (n+7)*(n-5) = 0.


The roots are n= -7  and  n= 5.   We want the positive value.


ANSWER.  The numbers are 5 and 6.


CHECK.  5^2 + 2*6 = 25 + 12 = 37.    ! correct !

Solved.


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Based on Edwin's note, someone can write a Master thesis or even a PhD dissertation in Math education :).

As an epigraph to such a Master thesis or PhD dissertation the person may write
that word problems in English often require a small auxiliary volume of explanations
of used terms and of what exactly means by the author, and what does not mean.



Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
There are multiple ways to interpret the wording of this problem.  Ikleyn
picked a way that did have a solution in positive integers, the first way
below.  However, the others are also valid ways to interpret the problem,
although some of the others likely do not have positive integer solutions.
Algebra problem-creators should be more careful to avoid ambiguous sentences.

1. Find two consecutive positive integers such that 
(the square of the first) increased by (2 times the second) is equal to 37. 

2. Find two consecutive positive integers such that 
[(the square of the first) increased by 2] times the second is equal to 37.

3. Find two consecutive positive integers such that 
[the square of (the first increased by 2)] times the second is equal to 37.

4. Find two consecutive positive integers such that
the square of [(the first increased by 2) times the second] is equal to 37.

5. Find two consecutive positive integers such that 
the square of [the first increased by (2 times the second)] is equal to 37.

6. Find two consecutive positive integers such that 
[the square of (the first increased by 2) times the second] is equal to 37.

Edwin