SOLUTION: 1.) Solve: sqrt x-3 = sqrt 2 - sqrt x for x 2.) Solve: sqrt 2y-6 = 3-y for y 3.) Solve: 5 - sqrt 20x + 4 greater than equal to -3 4.) The solution to sqrt 3x + 4 greater

Algebra ->  Square-cubic-other-roots -> SOLUTION: 1.) Solve: sqrt x-3 = sqrt 2 - sqrt x for x 2.) Solve: sqrt 2y-6 = 3-y for y 3.) Solve: 5 - sqrt 20x + 4 greater than equal to -3 4.) The solution to sqrt 3x + 4 greater       Log On


   

Question 996546: 1.) Solve: sqrt x-3 = sqrt 2 - sqrt x for x
2.) Solve: sqrt 2y-6 = 3-y for y
3.) Solve: 5 - sqrt 20x + 4 greater than equal to -3
4.) The solution to sqrt 3x + 4 greater than equal to 2 is x greater than or equal to 0 (true or false)
Answer by ikleyn(52798) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solve: sqrt x-3 = sqrt 2 - sqrt x for x
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sqrt%28x-3%29 = sqrt%282%29 - sqrt%28x%29.

Square both sides. You will get

x+-+3 = 2 - 2%2Asqrt%282%29%2Asqrt%28x%29 + x.

Simplify:

-3 - 2 = - 2%2Asqrt%282%29%2Asqrt%28x%29

5 = 2%2Asqrt%282%29%2Asqrt%28x%29.

Square both sides again:

25 = 8x.

x = 25%2F8.

Check:  Left side is   sqrt%28x-3%29 = sqrt%2825%2F8-3%29 = sqrt%28%2825-24%29%2F8%29 = sqrt%281%2F8%29 = 1%2F%282%2Asqrt%282%29%29.

            Right side is   sqrt%282%29 - sqrt%28x%29 = sqrt%282%29 - sqrt%2825%2F8%29 = sqrt%282%29 - %285%2F%282%2Asqrt%282%29%29%29 = %284-5%29%2F%282%2Asqrt%282%29%29 = -1%2F%282%2Asqrt%282%29%29.

The results for the left side and the right side are different.

It means that the original equation has no solutions.  (The solution that we obtained is the solution to the equation
with the squared left and right sides of the original equation,  but it doesn't satisfy the original equation.
Again,  it means that the original equation has no solutions).

The plots in the Figure below confirm this conclusion.




Plots: sqrt%28x-3%29  (red);  sqrt%282%29-sqrt%28x%29  (green);  sqrt%28x-3%29+-+%28sqrt%282%29-sqrt%28x%29%29  (blue).