SOLUTION: Solve. {{{sqrt(x-1)+sqrt(x-4)=sqrt(2x-1)}}}

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Question 124675: Solve.
sqrt%28x-1%29%2Bsqrt%28x-4%29=sqrt%282x-1%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%28x-1%29%2Bsqrt%28x-4%29=sqrt%282x-1%29 Start with the given equation


%28sqrt%28x-1%29%2Bsqrt%28x-4%29%29%5E2=%28sqrt%282x-1%29%29%5E2 Square both sides


%28sqrt%28x-1%29%2Bsqrt%28x-4%29%29%5E2=2x-1 Square the right side to remove the square root


sqrt%28x-1%29sqrt%28x-1%29%2B2sqrt%28x-1%29sqrt%28x-4%29%2Bsqrt%28x-4%29sqrt%28x-4%29=2x-1 Foil the left side

%28x-1%29%2B2sqrt%28%28x-1%29%28x-4%29%29%2B%28x-4%29=2x-1 Multiply


%28x-1%29%2B2sqrt%28x%5E2-5x%2B4%29%2B%28x-4%29=2x-1 Foil


2x-5%2B2sqrt%28x%5E2-5x%2B4%29=2x-1 Combine like terms


2sqrt%28x%5E2-5x%2B4%29=2x-1-2x%2B5 Subtract 2x from both sides. Add 5 to both sides.


2sqrt%28x%5E2-5x%2B4%29=4 Combine like terms


sqrt%28x%5E2-5x%2B4%29=2 Divide both sides by 2


x%5E2-5x%2B4=4 Square both sides


x%5E2-5x=0 Subtract 4 from both sides.


x%28x-5%29=0 Factor the left side


Now set each factor equal to zero:
x=0 or x-5=0

x=0 or x=5 Now solve for x in each case


So our possible answers are:
x=0 or x=5



Check:
Let's verify the first solution x=0

sqrt%28x-1%29%2Bsqrt%28x-4%29=sqrt%282x-1%29 Start with the given equation


sqrt%280-1%29%2Bsqrt%280-4%29=sqrt%282%280%29-1%29 Plug in x=0


sqrt%28-1%29%2Bsqrt%28-4%29=sqrt%28-1%29 Combine like terms


i%2B2i=i Simplify


3i=i Add. Since the two sides of the equation are not equal, this means that x=0 is not a solution.


-----------------

Let's verify the second solution x=5

sqrt%28x-1%29%2Bsqrt%28x-4%29=sqrt%282x-1%29 Start with the given equation


sqrt%285-1%29%2Bsqrt%285-4%29=sqrt%282%285%29-1%29 Plug in x=5


sqrt%284%29%2Bsqrt%281%29=sqrt%289%29 Combine like terms


2%2B1=3 Take the square root


3=3 Add. Since the two sides of the equation are equal, this verifies the solution


So the only solution is x=5