Question

Please put down the steps?

Answer 1

The answer is:  " x = 0, 1 " .
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Explanation:
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Given:
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   " √(x + 1)  − 1 = x "  ;   Solve for "x" ;

First, let us assume that "x ≥ -1 "

Add "1" to EACH SIDE of the equation:

 →  √(x + 1)  − 1  + 1 = x + 1  ; 

to get:  

 →  √(x + 1)  = x + 1 .

Now, "square" EACH side of the equation:

 →  [√(x + 1) ]²  = (x + 1 )² ;

to get: 

   x + 1 = (x + 1)² 

↔  (x + 1)²  = (x + 1) .


Expand the "left-hand side" of the equation:

 → (x + 1)² = (x + 1)(x +1) ;

Note: (a+b)(c+d) = ac +ad + bc + bd ;

As such:  (x + 1)(x + 1) = (x*x) + (x*1) +(1(x) + (1*1) ;
                                   =  x² + 1x + 1x + 1 ; 
                                   =  x²  + 2x  +  1 ;

Now, substitute this "expanded" value, and bring down the "right-hand side" of the equation; and rewrite the equation:
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   " (x + 1)² = (x + 1)(x +1) " ; 

  →  Rewrite as:  " x² + 2x + 1 = x + 1 " ;

 Subtract "x" ;  and subtract "1" ; from EACH SIDE of the equation:

               →   x² + 2x + 1 - x - 1 = x + 1 - x - 1 ; 

to get:    →   x² − x = 0  

Factor out an "x" on the "left-hand side" of the equation:
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         x² − x = x(x − 1) ;

         →  x (x − 1) = 0 ;

We have:  "x" and "(x − 1)" ; when either of these two multiplicands are equal to zero, then the "right-hand side of the equation equals "zero" .

So, one value of "x" is "0" .

The other value for "x" ;

→  x − 1 = 0 ;

Add "1" to each side of the equation:

  → x − 1 + 1 = 0 + 1 ; 

 →  x = 1 ; 
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So, the answers:

 " x = 0, 1 " .
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