`sqrt(4x)=x-3`
What are all values of x that satisfy the given equation?
I.1
II. 9

Options

A. I only
B. II only
C. I and II
D. Neither I nor II

Explanation

To solve the equation \( \sqrt{4x} = x - 3 \), we will follow these steps:

### Step 1: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ (\sqrt{4x})^2 = (x - 3)^2 \]
This simplifies to:
\[ 4x = (x - 3)(x - 3) \]

### Step 2: Expand the right side
Now, we expand the right side:
\[ 4x = x^2 - 6x + 9 \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to set it to zero:
\[ 0 = x^2 - 6x + 9 - 4x \]
This simplifies to:
\[ 0 = x^2 - 10x + 9 \]

### Step 4: Factor the quadratic equation
Now, we will factor the quadratic equation:
\[ 0 = (x - 1)(x - 9) \]

### Step 5: Solve for x
Setting each factor equal to zero gives us:
\[ x - 1 = 0 \quad \text{or} \quad x - 9 = 0 \]
Thus, we find:
\[ x = 1 \quad \text{or} \quad x = 9 \]

### Step 6: Check for extraneous solutions
We need to check both solutions in the original equation to ensure they are valid.

**For \( x = 1 \):**
\[ \sqrt{4 \cdot 1} = 1 - 3 \]
\[ \sqrt{4} = -2 \]
\[ 2 \neq -2 \quad \text{(not valid)} \]

**For \( x = 9 \):**
\[ \sqrt{4 \cdot 9} = 9 - 3 \]
\[ \sqrt{36} = 6 \]
\[ 6 = 6 \quad \text{(valid)} \]

### Conclusion
The only valid solution is \( x = 9 \).

### Final Answer
The values of \( x \) that satisfy the equation are:
\[ \text{Only } 9 \]

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