SOLUTION: A student is asked to solve √3x−4+√2+x= 2 and gives the following solution. Solution: {{{sqrt(3x-4)+sqrt(2+x)= 2}}} (3x−4)+(2+x)= 4 4x−2 =

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Question 375503: A student is asked to solve
√3x−4+√2+x= 2
and gives the following solution.
Solution:

(3x−4)+(2+x)= 4
4x−2 = 4
4x = 6
x = 3/2
Explain what the student did wrong and explain the correct procedure to solve the problem.

Answer by Edwin McCravy(20059)   (Show Source): You can put this solution on YOUR website!
A student is asked to solve

and gives the following solution.
Solution:

  <---Here is the error!  

This is the false assumption that "the square of the sum"
is the same as "the sum of the squares".  But it is not!
Never confuse these two!

Here is the correct solution:



Isolate one of the radicals. It doesn't matter which one. I'll isolate
the first one by subtracting the second one from both sides:



Indicate the squaring of both sides:



Simplify the left side by eliminating the square root and the 2 exponent.
Rewrite the right side as the binomial times itself:



Next we use FOIL on the right side:

     F   O    I    L
     |   |    |    | 


Combine the middle two radical terms on the right,
and simplify the first and last terms on the right:








Isolate the term with the radical on the left side



Divide every term by 2



Square both sides:










Get 0 on the left:



Swap sides:



This does not factor so we must use the quadratic formula:

 

 





Write 128 as 64*2



Simplify the radical:



Factor 2 out of the numerator:



Cancel the factors of 2:





We must check every radical equation for extraneous answers.

It is very difficult to check radicals inside radicals, and this
makes me wonder if you copied the problem right since this one
comes out so messy.  But I'll assume that you did.

We must check both solutions  and 
However to make things a little easier with these terrible answers,
we will use their decimal approximations,

Checking  



 








That does not check so  is an extraneous solution. 

----

Checking  



 








That is a very close check so  is 
the only solution to the original.

Edwin

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