Value of the roots of the quadratic equation, `x^(2)-x-6 = 0` are ...........

Answer

To find the roots of the quadratic equation \( x^2 - x - 6 = 0 \), we can follow these steps:

### Step 1: Write the equation
We start with the given quadratic equation:
\[ x^2 - x - 6 = 0 \]

### Step 2: Factor the quadratic
We need to factor the quadratic expression. We are looking for two numbers that multiply to \(-6\) (the constant term) and add up to \(-1\) (the coefficient of \(x\)).

The two numbers that satisfy these conditions are \(-3\) and \(2\) because:
- \(-3 \times 2 = -6\)
- \(-3 + 2 = -1\)

Thus, we can rewrite the equation as:
\[ x^2 - 3x + 2x - 6 = 0 \]

### Step 3: Group the terms
Now, we group the terms:
\[ (x^2 - 3x) + (2x - 6) = 0 \]

### Step 4: Factor by grouping
Next, we factor out the common terms in each group:
\[ x(x - 3) + 2(x - 3) = 0 \]

Now, we can factor out \((x - 3)\):
\[ (x - 3)(x + 2) = 0 \]

### Step 5: Set each factor to zero
Now, we set each factor equal to zero:
1. \(x - 3 = 0\) → \(x = 3\)
2. \(x + 2 = 0\) → \(x = -2\)

### Conclusion
The roots of the quadratic equation \(x^2 - x - 6 = 0\) are:
\[ x = 3 \quad \text{and} \quad x = -2 \]