Sum of solution of the equation `|x|^3-4|x|^2+3|x|=0` is

Options

A. 4
B. 3
C. 0
D. 1

Explanation

To find the sum of the solutions of the equation \( |x|^3 - 4|x|^2 + 3|x| = 0 \), we can follow these steps:

### Step 1: Substitute \( |x| \) with \( t \)
Let \( t = |x| \). Then, the equation becomes:
\[ t^3 - 4t^2 + 3t = 0 \]

### Step 2: Factor out \( t \)
We can factor out \( t \) from the equation:
\[ t(t^2 - 4t + 3) = 0 \]

### Step 3: Solve for \( t \)
Setting the factors equal to zero gives us:
1. \( t = 0 \)
2. \( t^2 - 4t + 3 = 0 \)

### Step 4: Solve the quadratic equation
To solve the quadratic equation \( t^2 - 4t + 3 = 0 \), we can use the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -4, c = 3 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 3 = 16 - 12 = 4 \]
Now substituting back into the quadratic formula:
\[ t = \frac{4 \pm \sqrt{4}}{2} = \frac{4 \pm 2}{2} \]
This gives us two solutions:
\[ t = \frac{6}{2} = 3 \quad \text{and} \quad t = \frac{2}{2} = 1 \]

### Step 5: List all possible values of \( x \)
Now we have the values for \( t \):
1. \( t = 0 \) implies \( |x| = 0 \) → \( x = 0 \)
2. \( t = 3 \) implies \( |x| = 3 \) → \( x = 3 \) or \( x = -3 \)
3. \( t = 1 \) implies \( |x| = 1 \) → \( x = 1 \) or \( x = -1 \)

### Step 6: Calculate the sum of all solutions
The solutions for \( x \) are \( 0, 3, -3, 1, -1 \). Now we can sum these values:
\[ 0 + 3 + (-3) + 1 + (-1) = 0 + 0 + 0 = 0 \]

### Conclusion
The sum of the solutions of the equation \( |x|^3 - 4|x|^2 + 3|x| = 0 \) is:
\[ \boxed{0} \]