How do you solve #sqrtx- sqrt(x-5)=1#?

Question

Eli Assouline · Accepted Answer

#x# = 9

#sqrt (x) -sqrt(x-5) = 1# equation 1 multiply both sides with #sqrt (x) +sqrt(x-5)# (#sqrt (x) -sqrt(x-5)) (sqrt (x) +sqrt(x-5))# = 1#(sqrt (x) +sqrt(x-5))# L H S is in the form of (a+b)(a-b) =# a^2 -b^2# #(sqrt(x))^2 - (sqrt(x-5))^2 # = #sqrt (x) +sqrt (x-5)# #x - (x-5) =sqrt (x) +sqrt(x-5)# #x-x+5 = (sqrt (x) +sqrt(x-5))# #5 = sqrt (x) +sqrt(x-5)# equation 2 Solve equation 1 and 2 for #x# #sqrt (x) -sqrt(x-5) = 1# equation 1 # sqrt (x) +sqrt(x-5) = 5# equation 2 Sum of equation 1 & 2 2#sqrt(x) = 6# #sqrt(x) = 6/2# = 3 Squaring on both side to get #x# #x = 9#

Hazel Anglin · Answer

undefined To solve the equation sqrt(x) - sqrt(x-5) = 1, you can follow these steps:

1. Start by isolating one of the square roots. Add sqrt(x-5) to both sides of the equation:
   sqrt(x) = sqrt(x-5) + 1

2. Square both sides of the equation to eliminate the square roots:
   (sqrt(x))^2 = (sqrt(x-5) + 1)^2

3. Simplify the equation:
   x = (sqrt(x-5) + 1)^2

4. Expand the right side of the equation:
   x = (sqrt(x-5))^2 + 2(sqrt(x-5)) + 1

5. Simplify further:
   x = x-5 + 2(sqrt(x-5)) + 1

6. Combine like terms:
   x = x + 2(sqrt(x-5)) - 4

7. Move all terms involving x to one side of the equation:
   x - x = 2(sqrt(x-5)) - 4

8. Simplify:
   0 = 2(sqrt(x-5)) - 4

9. Add 4 to both sides of the equation:
   4 = 2(sqrt(x-5))

10. Divide both sides of the equation by 2:
   2 = sqrt(x-5)

11. Square both sides of the equation to eliminate the square root:
   (2)^2 = (sqrt(x-5))^2

12. Simplify:
   4 = x-5

13. Add 5 to both sides of the equation:
   4 + 5 = x-5 + 5

14. Simplify:
   9 = x

Therefore, the solution to the equation sqrt(x) - sqrt(x-5) = 1 is x = 9.