Solution:

Given equation: √x^2-9x+18 = √x^2+2x-15 = √x^2-4x+3

Let's solve the equation step by step.

Step 1: Factorize the terms inside the square roots.

- √(x-3)(x-6) = √(x+5)(x-3) = √(x-1)(x-3)

Step 2: Equate the terms inside the square roots.

- (x-3)(x-6) = (x+5)(x-3) = (x-1)(x-3)

Step 3: Solve for x.

- x^2 - 9x + 18 = x^2 + 5x - 15 = x^2 - x - 3
- x^2 - x - 3 = 0
- Solving the quadratic equation, we get x = -1, 3.

Step 4: Check the validity of the roots.

- Plug in the values of x in the original equation to check if they satisfy the equation.
- For x = -1, the equation becomes 0 = 0, which is true.
- For x = 3, the equation becomes 0 = 0, which is also true.

Therefore, the following roots are obtained:

- x = -1
- x = 3

Explanation:

- The given equation is a complex equation with square roots and variables, which makes it difficult to solve.
- To solve the equation, we first need to factorize the terms inside the square roots.
- Then, we can equate the terms inside the square roots to get a simpler equation.
- Solving the quadratic equation, we get the values of x that satisfy the equation.
- Finally, we need to check the validity of the roots by plugging them into the original equation to see if they satisfy the equation.