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The equation given is:

\[
x - \sqrt{9 - 3x} = 0
\]

### Step 1: Isolate the square root term
We start by isolating the square root term:

\[
x = \sqrt{9 - 3x}
\]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:

\[
x^2 = 9 - 3x
\]

### Step 3: Rearrange into a standard quadratic form
Move all terms to one side of the equation to form a quadratic equation:

\[
x^2 + 3x - 9 = 0
\]

### Step 4: Solve the quadratic equation
We can now solve the quadratic equation using the quadratic formula. The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 + 3x - 9 = 0\), we have:
- \(a = 1\)
- \(b = 3\)
- \(c = -9\)

Substitute these values into the quadratic formula:

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

\[
x = \frac{-3 \pm \sqrt{9 + 36}}{2}
\]

\[
x = \frac{-3 \pm \sqrt{45}}{2}
\]

\[
x = \frac{-3 \pm 3\sqrt{5}}{2}
\]

Thus, the two possible solutions are:

\[
x = \frac{-3 + 3\sqrt{5}}{2} \quad \text{or} \quad x = \frac{-3 - 3\sqrt{5}}{2}
\]

### Step 5: Check for extraneous solutions
Since we squared both sides of the equation, we must check for extraneous solutions by substituting both values back into the original equation \(x - \sqrt{9 - 3x} = 0\).

Upon doing this check, we find that only the solution \(x = \frac{-3 + 3\sqrt{5}}{2}\) satisfies the original equation.

### Final Answer:
The real solution to the equation is:

\[
x = \frac{-3 + 3\sqrt{5}}{2}
\]

Would you like further details on any step, or do you have any questions?

Here are some related questions you might find useful:
1. How do you verify solutions when squaring both sides of an equation?
2. Can all square root equations have two real solutions?
3. How does the quadratic formula apply to different types of equations?
4. What are some examples of extraneous solutions and why do they occur?
5. How would you graph the function \( f(x) = x - \sqrt{9 - 3x} \)?

**Tip:** Always remember to check for extraneous solutions when solving equations involving square roots by substituting your results back into the original equation.

Find all real solutions of the equation: x - √(9 - 3x) = 0

This page covers the step-by-step solution to solve the equation x - √(9 - 3x) = 0 using algebraic manipulation and the quadratic formula. Explore important concepts such as square roots and quadratic equations, suitable for students in Grades 9-12.

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