The value of x satisfying `x^2 +5x + 6 = 0` is

Answer

To solve the quadratic equation \(x^2 + 5x + 6 = 0\), we can use two methods: factoring and the quadratic formula. Here, we will demonstrate both methods step by step.

### Method 1: Factoring

1. **Identify the equation**:
We start with the equation:
\[ x^2 + 5x + 6 = 0 \]

2. **Factor the quadratic**:
We need to find two numbers that multiply to \(6\) (the constant term) and add up to \(5\) (the coefficient of \(x\)). The numbers \(2\) and \(3\) satisfy this condition:
\[ 2 \times 3 = 6 \quad \text{and} \quad 2 + 3 = 5 \]

3. **Rewrite the equation**:
We can rewrite the quadratic equation as:
\[ (x + 2)(x + 3) = 0 \]

4. **Set each factor to zero**:
To find the values of \(x\), we set each factor equal to zero:
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

5. **Final solutions**:
The solutions to the equation are:
\[ x = -2 \quad \text{and} \quad x = -3 \]

### Method 2: Quadratic Formula

1. **Identify coefficients**:
For the equation \(x^2 + 5x + 6 = 0\), we identify \(a = 1\), \(b = 5\), and \(c = 6\).

2. **Use the quadratic formula**:
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

3. **Calculate the discriminant**:
First, we calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = 5^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1 \]

4. **Substitute into the formula**:
Now we substitute the values into the quadratic formula:
\[ x = \frac{-5 \pm \sqrt{1}}{2 \cdot 1} = \frac{-5 \pm 1}{2} \]

5. **Calculate the two possible values for \(x\)**:
- For the positive root:
\[ x = \frac{-5 + 1}{2} = \frac{-4}{2} = -2 \]
- For the negative root:
\[ x = \frac{-5 - 1}{2} = \frac{-6}{2} = -3 \]

6. **Final solutions**:
The solutions to the equation are:
\[ x = -2 \quad \text{and} \quad x = -3 \]

### Summary of Solutions
The values of \(x\) satisfying the equation \(x^2 + 5x + 6 = 0\) are:
\[ x = -2 \quad \text{and} \quad x = -3 \]