SOLUTION: square root of 3x-5 minus square root x+7 equals 2

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Question 30998: square root of 3x-5 minus square root x+7 equals 2
Found 2 solutions by longjonsilver, AnlytcPhil:
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!

+sqrt%283x-5%29+-+sqrt%28x%2B7%29+=+2+
+sqrt%283x-5%29+=+2+%2B+sqrt%28x%2B7%29+
+%28sqrt%283x-5%29%29%5E2+=+%282+%2B+sqrt%28x%2B7%29%29%5E2+
+3x-5+=+%282+%2B+sqrt%28x%2B7%29%29%282+%2B+sqrt%28x%2B7%29%29+
+3x-5+=+4+%2B+2sqrt%28x%2B7%29+%2B+2sqrt%28x%2B7%29+%2B+sqrt%28x%2B7%29sqrt%28x%2B7%29+
+3x-5+=+4+%2B+4sqrt%28x%2B7%29+%2B+%28x%2B7%29+
+3x-5+=+11+%2B+4sqrt%28x%2B7%29+%2B+x+
+2x-16+=+4sqrt%28x%2B7%29+
+%282x-16%29%2F4+=+sqrt%28x%2B7%29+
+%28x%2F2%29-4+=+sqrt%28x%2B7%29+
+%28%28x%2F2%29-4%29%5E2+=+%28sqrt%28x%2B7%29%29%5E2+
+%28%28x%2F2%29-4%29%28%28x%2F2%29-4%29+=+x%2B7+
+%28x%2F2%29%5E2+-4%28x%2F2%29+-+4%28x%2F2%29+%2B+16+=+x%2B7+
+%28x%5E2%2F4%29+-4x+%2B+16+=+x%2B7+
+%28x%5E2%2F4%29+-5x+%2B+16+=+7+
+%28x%5E2%2F4%29+-5x+%2B+9+=+0+
+x%5E2+-20x+%2B+36+=+0+
(x-18)(x-2) = 0

so x-18 = 0 or x-2=0
--> x=18 or x=2

CHECK:
when x=18: +sqrt%283x-5%29+-+sqrt%28x%2B7%29+ becomes
+sqrt%283%2818%29-5%29+-+sqrt%28%2818%29%2B7%29+
+sqrt%2854-5%29+-+sqrt%2818%2B7%29+
+sqrt%2849%29+-+sqrt%2825%29+
+7+-+5+
--> 2

also, when x=2: +sqrt%283x-5%29+-+sqrt%28x%2B7%29+ becomes
+sqrt%283%282%29-5%29+-+sqrt%282%2B7%29+
+sqrt%286-5%29+-+sqrt%282%2B7%29+
+sqrt%281%29+-+sqrt%289%29+

now...tricky: sqrt%281%29 is either +1 or -1. Similarly sqrt%289%29 is either +3 or -3. So, we get

-1 - (-3)
-1 + 3
--> 2

jon.

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
square root of 3x-5 minus square root x+7 equals 2
                    ____    ___
                   Ö3x-5 - Öx+7 = 2
                          ____            ___ 
Isolate the radical term Ö3x-5 by adding Öx+7 to both sides. 
                           ____        ___
                          Ö3x-5 = 2 + Öx+7 

Square both sides:

                          ____          ___
                       (Ö3x-5)² = (2 + Öx+7)² 

Careful here.  Since there is only ONE term in the parenthese on the left to be
squared, and since it is a square root, we can just take away the radical and
get 3x-5. But the right side id not so easy because there are TWO terms in the
parentheses.  We must write the parenbtheses on the right with TWO terms twice
and multiply using FOIL

                                        ___       ___
                           3x-5 = (2 + Öx+7)(2 + Öx+7)

To use FOIL on the right we have

F = 2×2 = 4
      ___
O = 2Öx+7
      ___
I = 2Öx+7
      ___
L = (Öx+7)² = x+7

So we have

                           3x - 5 = F + O + I + L
                                          ___     ___ 
                           3x - 5 = 4 + 2Öx+7 + 2Öx+7 + x + 7     
                                           ___
                           3x - 5 = 11 + 4Öx+7 + x
                                    ___ 
We again isolate the radical term 4Öx+7
                                       ___
                           2x - 16 = 4Öx+7

Evewry term can be divided by 2
                                       ___
                             x - 8 = 2Öx+7

Square both sides:
                                        _____
                          (x - 8)² = (2Öx + 7)²
                                         _____
                    (x - 8)(x - 8) = 2²(Öx + 7)²
             
                 x² - 8x - 8x + 64 = 4(x + 7)

                     x² - 16x + 64 = 4x + 28

                     x² - 20x + 36 = 0

                   (x - 18)(x - 2) = 0 

                   Answers: x = 18, x =2.
  
But we must ceck these in the original, because they both might be solutions,
one might be a solution and the other not, or there may be no solutions at all.
We substitute in the original equation:
                       ____    ___
                      Ö3x-5 - Öx+7 = 2  

                   _______    ____
                  Ö3(18)-5 - Ö18+7 = 2
                          __    __
                         Ö49 - Ö25 = 2

                             7 - 5 = 2

                                 2 = 2    

This checks so x = 18 is a solution
                       ____    ___
                      Ö3x-5 - Öx+7 = 2  

                     ______    ____
                    Ö3(2)-5 - Ö2+7 = 2
                            _    _
                           Ö1 - Ö9 = 2

                             1 - 3 = 2

                                -2 = 2

This does not check so x = 2 is NOT a solution                              

Edwin
AnlytcPhil@aol.com