Find all solutions in un-ordered integers ( a , b ) to 7 − a − b = 2 10 − 2 a b

Question

Find all solutions in un-ordered integers   (  a  ,  b  ) to   7  −  a  −  b  =  2      10    −  2      a    b

hitturn35 · Accepted Answer

Squaring both sides you get   (  7  −  a  −  b      )    2    −  40  −  4  a  b  =  −  8      10    a    b  . Since   10  a  b is an integer, and the square root of an integer is always an integer or irrational, we get that       10    a    b   is an integer. This shows that   a  b  =  10      r    2   for some   r  ∈      Z  . Then, the original equation implies   7  −  a  −  b  =  2      10    (  1  −      |    r      |    ). Since       10   is irrational and everything else in this last equation is an integer, we obtain   1  −      |    r      |    =  0. Thus,  r  =  +  1 or   −  1. Hence,   a  b  =  10. This restricts our choices for a,b because they have to be divisors of   10 so can only be from the set   {  1  ,  −  1  ,  2  ,  −  2  ,  5  ,  −  5  ,  10  ,  −  10  }.

Find all solutions in un-ordered integers (a,b) to 7−a−b=2sqrt 10−2sqrt ab.

Answered question

Answer & Explanation

New Questions in Algebra I