Product of roots of the equation `x - sqrt(3 x - 6) = 2` is

Options

A. 2
B. 5
C. 7
D. 10

Answer

To find the product of the roots of the equation \( x - \sqrt{3x - 6} = 2 \), we will follow these steps:

### Step 1: Rearranging the Equation
First, we rearrange the equation to isolate the square root:
\[ x - 2 = \sqrt{3x - 6} \]

**Hint:** Always isolate the square root when solving equations involving square roots.

### Step 2: Squaring Both Sides
Next, we square both sides to eliminate the square root:
\[ (x - 2)^2 = (\sqrt{3x - 6})^2 \]
This simplifies to:
\[ x^2 - 4x + 4 = 3x - 6 \]

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

### Step 3: Rearranging to Form a Quadratic Equation
Now, we rearrange the equation to bring all terms to one side:
\[ x^2 - 4x + 4 - 3x + 6 = 0 \]
This simplifies to:
\[ x^2 - 7x + 10 = 0 \]

**Hint:** Always combine like terms carefully when rearranging equations.

### Step 4: Factoring the Quadratic Equation
Next, we factor the quadratic equation:
\[ x^2 - 7x + 10 = (x - 5)(x - 2) = 0 \]

**Hint:** Look for two numbers that multiply to the constant term (10) and add to the coefficient of \( x \) (-7).

### Step 5: Finding the Roots
Setting each factor to zero gives us the roots:
1. \( x - 5 = 0 \) → \( x = 5 \)
2. \( x - 2 = 0 \) → \( x = 2 \)

**Hint:** Each factor gives a potential root; solve for \( x \) by setting each factor equal to zero.

### Step 6: Calculating the Product of the Roots
The product of the roots is:
\[ 5 \times 2 = 10 \]

**Hint:** The product of the roots can also be found using the formula \( \frac{c}{a} \) from the quadratic equation \( ax^2 + bx + c = 0 \).

### Conclusion
Thus, the product of the roots of the equation \( x - \sqrt{3x - 6} = 2 \) is \( \boxed{10} \).