Solving the Conundrum of Three Positive Integers

[aaronstonedd]

1. Three consecutive positive integers are such that the sum of the squares of the first two and the product of the other two is 46. Find the numbers. Variables: x. Three numbers: (x), (x + 1), (x + 2)



2. (I think, although I'm not sure.) x² + (x + 1)² + (x + 1)(x + 2) = 46



3.
x² + (x + 1)² + (x + 1)(x + 2) = 46
⇒ x² + x² + 1 + 2x + x² + 3x + 2 = 46
⇒ 3x² + 5x + 3 = 46
⇒ 3x² + 5x - 43 = 0
⇒ x² + (⁵/₃)x - ⁴³/₃ = ⁰/₃
⇒ x² + (2)(⁵/₆)(x) + (⁵/₃)² - (⁵/₃)² - ⁴³/3 = 0
⇒ (x + ⁵/₆)² = ⁴³/₃ + ²⁵/₃₆
⇒ (x + ⁵/₆)² = ^{516 + 25}/₃₆ = ⁵⁴¹/₃₆

Now this means x + ⁵/₆ is NOT a perfect square. And that means the three consecutive positive integers will also not be positive integers.

That is my predicament, of which I seek riddance.

 

[tiny-tim]

welcome to pf!

hi aaronstonedd! welcome to pf!

1. Three consecutive positive integers are such that the sum of the squares of the first two and the product of the other two is 46. Find the numbers. Variables: x. Three numbers: (x), (x + 1), (x + 2)



2. (I think, although I'm not sure.) x² + (x + 1)² + (x + 1)(x + 2) = 46

well, that can't be right …

the first two terms are one odd and one even, and the third term is even, so the total is odd, ≠ 46

perhaps it means the product of the outer two?

(and perhaps it means 96 ?)

 

[aaronstonedd]

So you're saying that even + odd + even = odd. That's right, because even + odd = odd. But it's also possible for three consecutive positive integers to be odd + even + odd which is equal to even!

Is the above argument valid?

 

[ehild]

Are you sure that the sum is not 45?

ehild

 

[aaronstonedd]

To ehild: Yes, I'm sure. Only the answer is given, from where I've got the question. The answer given is 4,5,6 although I'm not sure how that fits into the question's conditions.

The main problem I'm facing is interpreting the question and representing it mathematically.

 

[tiny-tim]

1. Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46. Find the numbers.

The answer given is 4,5,6

4*4 + 5*6 = 46

 

[ehild]

1. Three consecutive positive integers are such that the sum of the square of the first and the product of the other two is 46. Find the numbers.



4*4 + 5*6 = 46

Ingenious!

ehild

 

[aaronstonedd]

Thank you tiny-tim, for correcting the question. I've often noticed that if a question seems absurd or solving it gives odd results, then something in the question is probably wrong .

My query is solved. Thanks all!