Solving Equations:

To solve the given equations, we need to simplify them and then use algebraic techniques to isolate the variable (x) and solve for its value.

Simplifying Equations:

1. √ x^2 -9x +18:
We can factor the expression under the square root as (x-6)(x-3) and simplify the expression to √(x-6)(x-3).

2. √ x^2 +2x-15:
We can factor the expression under the square root as (x+5)(x-3) and simplify the expression to √(x+5)(x-3).

3. √ x^2 -4x +3:
We can factor the expression under the square root as (x-1)(x-3) and simplify the expression to √(x-1)(x-3).

Substituting Simplified Expressions:

Now, we can substitute the simplified expressions into the original equations:

1. √(x-6)(x-3) = √(x-1)(x-3) + 3

2. √(x+5)(x-3) = √(x-1)(x-3)

Isolating Variable:

We can now isolate the variable (x) in each equation by squaring both sides:

1. (x-6)(x-3) = [(x-1)(x-3) + 3]^2
2. (x+5)(x-3) = (x-1)(x-3)

Solving for Roots:

We can expand and simplify the equations to obtain:

1. x^2 - 9x + 15 = 0 OR x^2 - 7x + 7 = 0
Solving for x using the quadratic formula:
x = (9 ± √21)/2 OR x = (7 ± √29)/2

2. x^2 + 2x - 15 = 0
Solving for x using the quadratic formula:
x = -5 OR x = 3

Therefore, the roots obtained are:
-5, 3, (9 ± √21)/2, (7 ± √29)/2.