To solve the equation \(\sqrt{3x + 4} - \sqrt{x - 3} = 3\), let's follow these steps:

1. **Isolate one of the square roots:**

\[
\sqrt{3x + 4} = \sqrt{x - 3} + 3
\]

2. **Square both sides to eliminate the square root on the left-hand side:**

\[
(\sqrt{3x + 4})^2 = (\sqrt{x - 3} + 3)^2
\]

\[
3x + 4 = (x - 3) + 6\sqrt{x - 3} + 9
\]

\[
3x + 4 = x - 3 + 6\sqrt{x - 3} + 9
\]

\[
3x + 4 = x + 6\sqrt{x - 3} + 6
\]

3. **Isolate the term with the square root:**

\[
3x + 4 - x - 6 = 6\sqrt{x - 3}
\]

\[
2x - 2 = 6\sqrt{x - 3}
\]

\[
\frac{2x - 2}{6} = \sqrt{x - 3}
\]

\[
\frac{x - 1}{3} = \sqrt{x - 3}
\]

4. **Square both sides again to eliminate the square root:**

\[
\left(\frac{x - 1}{3}\right)^2 = (\sqrt{x - 3})^2
\]

\[
\frac{(x - 1)^2}{9} = x - 3
\]

\[
(x - 1)^2 = 9(x - 3)
\]

5. **Expand and simplify:**

\[
x^2 - 2x + 1 = 9x - 27
\]

\[
x^2 - 2x + 1 - 9x + 27 = 0
\]

\[
x^2 - 11x + 28 = 0
\]

6. **Solve the quadratic equation:**

\[
x^2 - 11x + 28 = 0
\]

This can be factored as:

\[
(x - 4)(x - 7) = 0
\]

So, the solutions are:

\[
x = 4 \quad \text{or} \quad x = 7
\]

7. **Verify the solutions in the original equation:**

For \( x = 4 \):

\[
\sqrt{3(4) + 4} - \sqrt{4 - 3} = \sqrt{12 + 4} - \sqrt{1} = \sqrt{16} - 1 = 4 - 1 = 3 \quad \text{(Valid)}
\]

For \( x = 7 \):

\[
\sqrt{3(7) + 4} - \sqrt{7 - 3} = \sqrt{21 + 4} - \sqrt{4} = \sqrt{25} - 2 = 5 - 2 = 3 \quad \text{(Valid)}
\]

So, both solutions \( x = 4 \) and \( x = 7 \) are valid.

**Final Answer:**

\[
x = 4 \quad \text{or} \quad x = 7
\]

Would you like more details or have any questions?

Here are some related questions:

1. How do you solve equations involving two square roots?
2. What are some common techniques for solving quadratic equations?
3. How can you check if your solutions to an equation are correct?
4. What is the general method for solving radical equations?
5. How do you handle extraneous solutions in equations involving square roots?
6. What are the properties of square root functions?
7. Can you explain the process of isolating terms in an equation?
8. How do you simplify expressions with radicals?

**Tip:** Always check your solutions in the original equation to ensure they are valid and to catch any extraneous solutions.

Learn how to solve the equation √(3x + 4) - √(x - 3) = 3 with a detailed step-by-step solution. This problem covers key concepts including square roots and quadratic equations. Suitable for advanced high school students (Grades 10-12).

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