The product of the roots of the equation `x|x|-5x-6=0` is equal to

Options

A. 36
B. `-36`
C. `-18`
D. 6

Answer

To find the product of the roots of the equation \( x|x| - 5x - 6 = 0 \), we will analyze the equation by considering two cases based on the definition of the absolute value function.

### Step 1: Break down the equation based on the sign of \( x \)

The absolute value function \( |x| \) behaves differently based on whether \( x \) is positive or negative. Therefore, we will consider two cases:

- **Case 1**: \( x \geq 0 \)
- **Case 2**: \( x < 0 \)

### Step 2: Solve Case 1: \( x \geq 0 \)

In this case, \( |x| = x \). Thus, the equation becomes:
\[ x^2 - 5x - 6 = 0 \]
Now, we will use the quadratic formula to find the roots:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -5, c = -6 \):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{5 \pm \sqrt{25 + 24}}{2} = \frac{5 \pm \sqrt{49}}{2} = \frac{5 \pm 7}{2} \]
This gives us two roots:
\[ x_1 = \frac{12}{2} = 6 \quad \text{and} \quad x_2 = \frac{-2}{2} = -1 \]
Since we are in the case where \( x \geq 0 \), we only take \( x_1 = 6 \).

### Step 3: Solve Case 2: \( x < 0 \)

In this case, \( |x| = -x \). Thus, the equation becomes:
\[ -x^2 - 5x - 6 = 0 \]
Multiplying through by -1 gives:
\[ x^2 + 5x + 6 = 0 \]
Again, we will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 5, c = 6 \):
\[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{-5 \pm \sqrt{25 - 24}}{2} = \frac{-5 \pm \sqrt{1}}{2} = \frac{-5 \pm 1}{2} \]
This gives us two roots:
\[ x_3 = \frac{-4}{2} = -2 \quad \text{and} \quad x_4 = \frac{-6}{2} = -3 \]
Both roots \( x_3 = -2 \) and \( x_4 = -3 \) are valid since we are in the case where \( x < 0 \).

### Step 4: Collect all roots

From both cases, we have the roots:
- From Case 1: \( 6 \)
- From Case 2: \( -2, -3 \)

### Step 5: Calculate the product of the roots

The product of the roots is:
\[ P = 6 \times (-2) \times (-3) \]
Calculating this:
\[ P = 6 \times 2 \times 3 = 36 \]

### Final Answer

The product of the roots of the equation \( x|x| - 5x - 6 = 0 \) is equal to \( 36 \).

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