SOLUTION: square root of 4x+5-square root of 3x-6=8

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Question 282020: square root of 4x+5-square root of 3x-6=8
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
 
sqrt%284x%2B5%29-sqrt%283x-6%29=8

Let A=sqrt%284x%2B5%29 and B=sqrt%283x-6%29

A-B=8

Solve for one of the letters, say A:

A=8%2BB

Square both sides:

A%5E2=%288%2BB%29%5E2

A%5E2=64%2B16B%2BB%5E2

Replace A%5E2 by A=4x%2B5 and B%5E2 by 3x-6

4x%2B5=64%2B16B%2B3x-6

Solve for 16B:

4x%2B5=58%2B16B%2B3x

x%2B5=58%2B16B

x-53=16B

Square both sides:

x%5E2-106x%2B2809=256B%5E2

Replace B%5E2 by 3x-6

x%5E2-106x%2B2809=256%283x-6%29

x%5E2-106x%2B2809=768x-1536

x%5E2-874x%2B4345=0

Factor the left side:

%28x-869%29%28x-5%29=0

Solutions are x=869 and x=5

We must check even root radical equations for extraneous solutions:

Substitute x=869 into original equation:

sqrt%284x%2B5%29-sqrt%283x-6%29=8

sqrt%284%28869%29%2B5%29-sqrt%283%28869%29-6%29=8

sqrt%283481%29-sqrt%282601%29=8

59-51=8

8=8

That is true so x=869 is a solution to the
original equation.

Substitute x=5 into original equation:

sqrt%284x%2B5%29-sqrt%283x-6%29=8

sqrt%284%285%29%2B5%29-sqrt%283%285%29-6%29=8

sqrt%2825%29-sqrt%289%29=8

5-3=8

2=8

That is false so x=5 is not a solution to the
original equation.

So there is one solution x=869

Edwin