Solve the equation:  sqrt(2x - 5) - sqrt(x - 3) = 1 If there is more than one solution, enter your solutions separated by a comma. If ther is no solution, enter "none".

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baxthum8's profile pic

baxthum8 | High School Teacher | (Level 3) Associate Educator

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Solve `sqrt(2x - 5) -sqrt(x-3) = 1`

`(sqrt(2x - 5) -sqrt(x-3))^2 = 1^2`

`2x - 5 - 2sqrt(2x^2 - 11x + 15) + x - 3 = 1`

`3x - 8 - 2sqrt(2x^2 - 11x + 15) = 1`

`3x - 9 = 2sqrt(2x^2 - 11x + 15)`

`(3x - 9)^2 = (2sqrt(2x^2 - 11x + 15))^2`

`9x^2 - 54x + 81 = 4(2x^2 -11x + 15)`

`9x^2 - 54x + 81 = 8x^2 - 44x + 60`

`x^2 - 10x + 21 = 0`

`(x - 3) (x - 7) = 0`

`x = 3 and x = 7`

The solutions to the equation are 3 and 7, or `x = 3, 7`

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aruv | High School Teacher | (Level 2) Valedictorian

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x=3 will not satisfy given equation, so it can not be solution.

The problem has only one solution x=7

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tonys538 | Student, Undergraduate | (Level 1) Valedictorian

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The solution of `sqrt(2x - 5) - sqrt(x - 3) = 1` has to be determined.

`sqrt(2x - 5) - sqrt(x - 3) = 1`

Take the square of both the sides

`(sqrt(2x - 5) - sqrt(x - 3))^2 = 1^2`

`2x - 5 + x - 3 - 2*sqrt(2x-5)*sqrt(x-3) = 1`

`3x - 2*sqrt(2x-5)*sqrt(x-3) = 1 + 8`

`- 2*sqrt(2x-5)*sqrt(x-3) = 9 - 3x`

Take the square of the two sides again

4*(2x - 5)*(x - 3) = 81 + 9x^2 - 54x

8x^2 - 24x - 20x + 60 = 81 + 9x^2 - 54x

10x = x^2 + 21

x^2 - 10x + 21 = 0

x^2 - 3x - 7x + 21 = 0

x(x - 3) - 7(x - 3) = 0

(x - 3)(x - 7) = 0

x = 3 and x = 7

The solution of the equation is x = 3 and x = 7

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