The solution of the equation`(3|x|-3)^2 = |x|+7` which belongs to the domain of `sqrt(x(x-3)` are given by

Options

A. F F T
B. T T F
C. T F F
D. T T T

Answer

To solve the equation \((3|x| - 3)^2 = |x| + 7\) under the domain of \(\sqrt{x(x-3)}\), we will follow these steps:

### Step 1: Determine the Domain
The expression \(\sqrt{x(x-3)}\) is defined when \(x(x-3) \geq 0\). This means:
1. \(x \geq 0\) and \(x - 3 \geq 0\) (which gives \(x \geq 3\))
2. or \(x \leq 0\) and \(x - 3 \leq 0\) (which gives \(x \leq 0\))

Thus, the domain is \(x \leq 0\) or \(x \geq 3\).

### Step 2: Solve the Equation for \(x \leq 0\)
For \(x \leq 0\), we have \(|x| = -x\). Substitute this into the equation:
\[ (3(-x) - 3)^2 = -x + 7 \]
This simplifies to:
\[ (-3x - 3)^2 = -x + 7 \]
Expanding the left side:
\[ (9x^2 + 18x + 9) = -x + 7 \]
Rearranging gives:
\[ 9x^2 + 19x + 2 = 0 \]

### Step 3: Factor the Quadratic Equation
Now we need to factor the quadratic equation \(9x^2 + 19x + 2 = 0\). We can factor it as:
\[ (9x + 1)(x + 2) = 0 \]
Setting each factor to zero gives us:
1. \(9x + 1 = 0 \Rightarrow x = -\frac{1}{9}\)
2. \(x + 2 = 0 \Rightarrow x = -2\)

### Step 4: Check Validity of Solutions
Now we need to check if these solutions belong to the domain:
- For \(x = -\frac{1}{9}\): This is not in the domain \(x \leq 0\).
- For \(x = -2\): This is in the domain \(x \leq 0\).

Thus, the only valid solution from this case is \(x = -2\).

### Step 5: Solve the Equation for \(x \geq 3\)
For \(x \geq 3\), we have \(|x| = x\). Substitute this into the equation:
\[ (3x - 3)^2 = x + 7 \]
This simplifies to:
\[ (3x - 3)^2 = x + 7 \]
Expanding the left side:
\[ (9x^2 - 18x + 9) = x + 7 \]
Rearranging gives:
\[ 9x^2 - 19x + 2 = 0 \]

### Step 6: Factor the Quadratic Equation
Now we need to factor the quadratic equation \(9x^2 - 19x + 2 = 0\). We can factor it as:
\[ (9x - 1)(x - 2) = 0 \]
Setting each factor to zero gives us:
1. \(9x - 1 = 0 \Rightarrow x = \frac{1}{9}\) (not valid since \(x \geq 3\))
2. \(x - 2 = 0 \Rightarrow x = 2\) (not valid since \(x \geq 3\))

### Conclusion
The only solution that belongs to the domain of \(\sqrt{x(x-3)}\) is:
\[ \boxed{-2} \]