To solve the equation \(\sqrt{x + 3} - 3 = \sqrt{2 - x}\), we will first isolate one of the square roots and then square both sides of the equation to eliminate the square roots.

1. Isolate \(\sqrt{x + 3}\):
\[
\sqrt{x + 3} = \sqrt{2 - x} + 3
\]

2. Square both sides of the equation:
\[
(\sqrt{x + 3})^2 = (\sqrt{2 - x} + 3)^2
\]
\[
x + 3 = (2 - x) + 6\sqrt{2 - x} + 9
\]

3. Simplify the equation:
\[
x + 3 = 11 - x + 6\sqrt{2 - x}
\]

4. Isolate the square root term:
\[
x + x + 3 - 11 = 6\sqrt{2 - x}
\]
\[
2x - 8 = 6\sqrt{2 - x}
\]

5. Simplify further:
\[
\frac{2x - 8}{6} = \sqrt{2 - x}
\]
\[
\frac{x - 4}{3} = \sqrt{2 - x}
\]

6. Square both sides again:
\[
\left(\frac{x - 4}{3}\right)^2 = (2 - x)
\]
\[
\frac{(x - 4)^2}{9} = 2 - x
\]
\[
(x - 4)^2 = 9(2 - x)
\]
\[
(x - 4)^2 = 18 - 9x
\]

7. Expand and simplify:
\[
x^2 - 8x + 16 = 18 - 9x
\]
\[
x^2 - 8x + 9x + 16 - 18 = 0
\]
\[
x^2 + x - 2 = 0
\]

8. Solve the quadratic equation:
\[
x^2 + x - 2 = 0
\]

To solve the quadratic equation, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 1\), and \(c = -2\):

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-2)}}{2(1)}
\]
\[
x = \frac{-1 \pm \sqrt{1 + 8}}{2}
\]
\[
x = \frac{-1 \pm \sqrt{9}}{2}
\]
\[
x = \frac{-1 \pm 3}{2}
\]

So, the solutions are:
\[
x = \frac{-1 + 3}{2} = 1
\]
\[
x = \frac{-1 - 3}{2} = -2
\]

9. Check both solutions in the original equation to ensure they are valid:
For \(x = 1\):
\[
\sqrt{1 + 3} - 3 = \sqrt{2 - 1}
\]
\[
\sqrt{4} - 3 = \sqrt{1}
\]
\[
2 - 3 = 1 \quad \text{(False)}
\]

For \(x = -2\):
\[
\sqrt{-2 + 3} - 3 = \sqrt{2 - (-2)}
\]
\[
\sqrt{1} - 3 = \sqrt{4}
\]
\[
1 - 3 = 2 \quad \text{(False)}
\]

Both solutions do not satisfy the original equation. Therefore, there are no real solutions to the equation \(\sqrt{x + 3} - 3 = \sqrt{2 - x}\).

Would you like more details or have any questions about the solution process?

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Here are 8 related questions that might help expand your understanding:

1. What are the general steps to solve equations involving square roots?
2. How do you isolate a square root in an equation?
3. What are some common mistakes to avoid when solving radical equations?
4. How can you verify if a solution is valid for a given equation?
5. What is the importance of squaring both sides when dealing with square root equations?
6. Can there be extraneous solutions when solving radical equations? Why?
7. How do you solve quadratic equations using the quadratic formula?
8. Why is it necessary to check potential solutions in the original equation?

**Tip:** Always check your potential solutions in the original equation when dealing with square roots to ensure they do not introduce extraneous solutions.

Learn how to solve the equation sqrt(x + 3) - 3 = sqrt(2 - x) step-by-step with detailed explanations and solutions. This problem involves equations with square roots and quadratic formula applications. Suitable for advanced high school students.

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