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Find all x that satisfy the equation

\[ \sqrt{2x+1} + \sqrt{x-3} = 2\sqrt x.\]

 Jan 22, 2021
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First, square both sides

 

2x + 1   + 2sqrt [ ( 2x + 1)(x  - 3) ]  + x  - 3 =   4x     simplify

 

3x - 2 +  2sqrt [ ( 2x + 1)(x - 3)  ]  =  4x

 

2sqrt  [  ( 2x + 1) ( x  -3) ]   =  x + 2        square both sides again

 

4  ( 2x + 1) ( x - 3)  = x^2 + 4x + 4

 

4 ( 2x^2 -5x - 3)  = x^2  + 4x + 4

 

8x^ 2  - 20x - 12  = x^2  + 4x + 4

 

7x^2 - 24x  - 16  =   0      factor as

 

(7x  + 4 ) ( x  -4  )   =  0

 

The first factor  set to 0 and solved for x   gives  x =  -4/7

This is no good  because  we get  negatives  under  all the radicals with this solution

 

The second factor set  to 0 and solved for x gives us the  solution   x  = 4

 

cool cool cool

 Jan 22, 2021
edited by CPhill  Jan 22, 2021

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