SOLUTION: Solve 2[√(4x+5)] - [√(3x+1)] = 6, where x is an integer.

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Question 1210433: Solve 2[√(4x+5)] - [√(3x+1)] = 6, where x is an integer.
Answer by mccravyedwin(417) About Me  (Show Source):
You can put this solution on YOUR website!

2%28sqrt%284x%2B5%29%29+-+%28sqrt%283x%2B1%29%29+=+6

Isolate a radical term:

2sqrt%284x%2B5%29+=+6+%2B+sqrt%283x%2B1%29

Square both sides:

%282sqrt%284x%2B5%29%5E%22%22%29%5E2+=+%286+%2B+sqrt%283x%2B1%29%5E%22%22%29%5E2

Simplify:

4%284x%2B5%29+=+36+%2B+12sqrt%283x%2B1%29+%2B+3x%2B1

16x%2B20+=+37+%2B+12sqrt%283x%2B1%29+%2B+3x

16x%2B20+=+37+%2B+12sqrt%283x%2B1%29+%2B+3x

Isolate the radical term:

13x-17+=+12sqrt%283x%2B1%29

Square both sides:

%2813x-17%29%5E2+=+12sqrt%283x%2B1%29 

169x%5E2-442x%2B289+=+144%283x%2B1%29

169x%5E2-442x%2B289+=+432x%2B144%29

169x%5E2-874x%2B145=0

%28169x-29%29%28x-5%29+=+0

 169x-29=0;  x-5=0
 169x = 29;    x=5
 x=29/169;

Since x is an integer, the answer is likely x = 5.

But since we squared both sides, we must check for extraneous roots.

2%28sqrt%284x%2B5%29%29+-+%28sqrt%283x%2B1%29%29+=+6
2%28sqrt%284%2A5%2B5%29%29+-+%28sqrt%283%2A5%2B1%29%29+=+6
2%28sqrt%2825%29%29+-+%28sqrt%2816%29%29+=+6
2%285%29+-+4+=+6
10-4=6
6=6

It checks, so the answer is x = 5.

Edwin