Final answer:
To solve the radical equation √2x-4 -√x-4 =2, add √2x-4 to both sides, square both sides of the equation, simplify, and solve for x. The solution is x = 14.
Step-by-step explanation:
To solve the radical equation √(2x-4) - √(x-4) = 2, we'll first isolate one of the square roots and then square both sides to eliminate the radical. Here are the detailed steps:
- Isolate one of the square roots: √(2x-4) = √(x-4) + 2
- Square both sides of the equation: (2x-4) = (√(x-4) + 2)^2
- Expand the right side: 2x - 4 = (x-4) + 4√(x-4) + 4
- Isolate the remaining square root: 4√(x-4) = 2x - x - 4 + 4
- Simplify: 4√(x-4) = x
- Square both sides again: 16(x-4) = x^2
- Expand and move all terms to one side: x^2 - 16x + 64 = 0
- Solve the quadratic equation using factoring or the quadratic formula
The solution is x = 14.