Let's go through and solve each of these equations step-by-step.

(a) $\sqrt{2x + 7} = \sqrt{x + 3} + 1$

1. Isolate the square root term on one side: $\sqrt{2x + 7} - \sqrt{x + 3} = 1$

2. Square both sides to eliminate square roots: $(\sqrt{2x + 7} - \sqrt{x + 3})^2 = 1^2$ Expanding both sides: $(2x + 7) - 2\sqrt{(2x + 7)(x + 3)} + (x + 3) = 1$ Simplify: $3x + 10 - 2\sqrt{(2x + 7)(x + 3)} = 1$ $3x + 9 = 2\sqrt{(2x + 7)(x + 3)}$

3. Square both sides again: $(3x + 9)^2 = [2\sqrt{(2x + 7)(x + 3)}]^2$ Simplify: $(3x + 9)^2 = 4(2x + 7)(x + 3)$ Solve the resulting quadratic equation.

(b) $\sqrt{x} = \sqrt{x - 5} + 1$

1. Isolate the square root term: $\sqrt{x} - \sqrt{x - 5} = 1$

2. Square both sides: $(\sqrt{x} - \sqrt{x - 5})^2 = 1^2$ Expand: $x - 2\sqrt{x(x - 5)} + (x - 5) = 1$ Simplify: $2x - 5 = 2\sqrt{x(x - 5)}$

3. Square both sides again and solve for $x$.

(c) $\sqrt{3x + 4} - \sqrt{2x + 1} = 1$

1. Isolate the square root: $\sqrt{3x + 4} = \sqrt{2x + 1} + 1$

2. Square both sides: $(3x + 4) = (\sqrt{2x + 1} + 1)^2$ Expand: $3x + 4 = (2x + 1) + 2\sqrt{2x + 1} + 1$ Simplify and solve the resulting equation.

(d) $\sqrt{x + 1} - \sqrt{x} = 1$

1. Isolate the square root: $\sqrt{x + 1} = \sqrt{x} + 1$

2. Square both sides: $(x + 1) = (\sqrt{x} + 1)^2$ Expand and solve.

(e) $\sqrt{x + 1} + \sqrt{2x + 3} = 5$

1. Isolate one square root: $\sqrt{x + 1} = 5 - \sqrt{2x + 3}$

2. Square both sides and simplify.

(f) $\sqrt{16 - 2x} = 2 + \sqrt{36 + 6x}$

1. Isolate one square root: $\sqrt{16 - 2x} - \sqrt{36 + 6x} = 2$

2. Square both sides and simplify.

Would you like me to go deeper into the steps for any particular equation?

Follow-up Questions:

1. What methods can you use to simplify square root equations?
2. How do you check for extraneous solutions in square root equations?
3. What is the significance of squaring both sides in these problems?
4. Are there alternative strategies for solving square root equations without squaring both sides?
5. How do you verify if your solution is valid after solving the equations?

Tip:

When squaring both sides in square root equations, always check for extraneous solutions by substituting the solutions back into the original equation.