Re: Algebric problem

On 6/26/2024 10:17 AM, Richard Hachel wrote:

One commenter posed the problem:
 sqrt(3x+7)+sqrt(x+2)=1
 On the French forum,
mathematicians had some problems with this.
 However,
the solution is very simple,
and requires three lines of calculations
on a piece of paper.

Is the problem that
some of the possible solutions presented by
three lines of calculations
do not produce equality in the original equation?
Squaring can introduce more possible solutions.
for each x such that  √(3x+7)+√(x+2)=1
also  √(3x+7)=1-√(x+2)
for each x such that  √(3x+7)=1-√(x+2)
also  3x+7=1-2√(x+2)+x+2
but also
for each x such that  -√(3x+7)=1-√(x+2)
also  3x+7=1-2√(x+2)+x+2
for each x such that
  √(3x+7)=1-√(x+2) or
  -√(3x+7)=1-√(x+2)
also
  3x+7=1-2√(x+2)+x+2  and
  x+2=-√(x+2)
for each x such that
  √(3x+7)=1-√(x+2) or
  -√(3x+7)=1-√(x+2) or
  x+2=-√(x+2) or
  x+2=√(x+2)
also
  (x+2)²=x+2
  (x+1)(x+2)=0
  x=-1 or x=-2
Only -1 and -2 _can be_ a solution.
√(3⋅-1+7)+√(-1+2)=√4+√1≠1
√(3⋅-2+7)+√(-2+2)=√1+√0=1
Only -2 _is_ a solution.

 R.H.