The product of real roots of the equation `|3x-4|^(2)-3|3x-4|+2=0` is

Options

A. `(10)/(9)`
B. `(20)/(9)`
C. `(1)/(3)`
D. none of these

Answer

To solve the equation \(|3x-4|^2 - 3|3x-4| + 2 = 0\) and find the product of its real roots, we can follow these steps:

### Step 1: Substitute \( t = |3x - 4| \)

Let \( t = |3x - 4| \). Then the equation transforms to:

\[ t^2 - 3t + 2 = 0 \]

### Step 2: Factor the quadratic equation

We can factor the quadratic equation:

\[ t^2 - 3t + 2 = (t - 1)(t - 2) = 0 \]

### Step 3: Solve for \( t \)

Setting each factor to zero gives us:

\[ t - 1 = 0 \quad \Rightarrow \quad t = 1 \]
\[ t - 2 = 0 \quad \Rightarrow \quad t = 2 \]

### Step 4: Solve for \( x \) using \( t = |3x - 4| \)

Now we will solve for \( x \) in both cases.

#### Case 1: \( t = 1 \)

\[ |3x - 4| = 1 \]

This gives us two equations:

1. \( 3x - 4 = 1 \)
\[ 3x = 5 \quad \Rightarrow \quad x = \frac{5}{3} \]

2. \( 3x - 4 = -1 \)
\[ 3x = 3 \quad \Rightarrow \quad x = 1 \]

#### Case 2: \( t = 2 \)

\[ |3x - 4| = 2 \]

This gives us two more equations:

1. \( 3x - 4 = 2 \)
\[ 3x = 6 \quad \Rightarrow \quad x = 2 \]

2. \( 3x - 4 = -2 \)
\[ 3x = 2 \quad \Rightarrow \quad x = \frac{2}{3} \]

### Step 5: List all real roots

The real roots we found are:

1. \( x = \frac{5}{3} \)
2. \( x = 1 \)
3. \( x = 2 \)
4. \( x = \frac{2}{3} \)

### Step 6: Calculate the product of the real roots

Now we calculate the product of all the real roots:

\[ \text{Product} = \left(\frac{5}{3}\right) \times 1 \times 2 \times \left(\frac{2}{3}\right) \]

Calculating this step-by-step:

1. \( \frac{5}{3} \times 1 = \frac{5}{3} \)
2. \( \frac{5}{3} \times 2 = \frac{10}{3} \)
3. \( \frac{10}{3} \times \frac{2}{3} = \frac{20}{9} \)

### Final Answer

The product of the real roots is:

\[ \frac{20}{9} \]

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