Let `f(x)=sqrt(x^(2)-4x)` and g(x) = 3x . The sum of all values for which f(x) = g(x) is

Options

A. `-8.5`
B. 0
C. 8
D. 9.4

Explanation

To solve the equation \( f(x) = g(x) \) where \( f(x) = \sqrt{x^2 - 4x} \) and \( g(x) = 3x \), we will follow these steps:

### Step 1: Set the functions equal to each other
We start by equating \( f(x) \) and \( g(x) \):
\[ \sqrt{x^2 - 4x} = 3x \]

**Hint:** Always start by setting the two functions equal to each other when solving for their intersection points.

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ (\sqrt{x^2 - 4x})^2 = (3x)^2 \]
This simplifies to:
\[ x^2 - 4x = 9x^2 \]

**Hint:** Squaring both sides is a common technique to eliminate square roots, but remember to check for extraneous solutions later.

### Step 3: Rearrange the equation
Now, we rearrange the equation to bring all terms to one side:
\[ x^2 - 9x^2 - 4x = 0 \]
This simplifies to:
\[ -8x^2 - 4x = 0 \]
Factoring out the common term:
\[ -4x(2x + 1) = 0 \]

**Hint:** Rearranging helps in identifying common factors which can be factored out easily.

### Step 4: Solve for \( x \)
Setting each factor to zero gives us:
1. \( -4x = 0 \) which leads to \( x = 0 \)
2. \( 2x + 1 = 0 \) which leads to \( x = -\frac{1}{2} \)

**Hint:** Always set each factor equal to zero to find all possible solutions.

### Step 5: Check for extraneous solutions
Since \( f(x) = \sqrt{x^2 - 4x} \), we need to ensure that the values of \( x \) do not make the expression under the square root negative.

For \( x = 0 \):
\[ f(0) = \sqrt{0^2 - 4(0)} = 0 \quad \text{and} \quad g(0) = 3(0) = 0 \]
This is valid.

For \( x = -\frac{1}{2} \):
\[ f\left(-\frac{1}{2}\right) = \sqrt{\left(-\frac{1}{2}\right)^2 - 4\left(-\frac{1}{2}\right)} = \sqrt{\frac{1}{4} + 2} = \sqrt{\frac{9}{4}} = \frac{3}{2} \]
And:
\[ g\left(-\frac{1}{2}\right) = 3\left(-\frac{1}{2}\right) = -\frac{3}{2} \]
Since \( f\left(-\frac{1}{2}\right) \neq g\left(-\frac{1}{2}\right) \), this value is extraneous.

**Hint:** Always check your solutions in the original equations to ensure they are valid.

### Step 6: Sum the valid solutions
The only valid solution is \( x = 0 \). The sum of all valid solutions is:
\[ 0 \]

### Final Answer
Thus, the sum of all values for which \( f(x) = g(x) \) is:
\[ \boxed{0} \]