Question

Solve the quadratic equation x^2 + 5x + 6 = 0 and show the steps to find the values of x.

Answer 1

To solve the quadratic equation \( x^2 + 5x + 6 = 0 \), we can use the factoring method.

### Step 1: Factor the quadratic expression
We are looking for two numbers that multiply to \( 6 \) (the constant term) and add to \( 5 \) (the coefficient of the linear term).

The two numbers that satisfy these conditions are \( 2 \) and \( 3 \), since:
- \( 2 \times 3 = 6 \)
- \( 2 + 3 = 5 \)

### Step 2: Write the equation in factored form
Using the numbers we found, we can factor the quadratic:
\[
x^2 + 5x + 6 = (x + 2)(x + 3) = 0
\]

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

### Step 4: Solve for \( x \)
From \( x + 2 = 0 \):
\[
x = -2
\]

From \( x + 3 = 0 \):
\[
x = -3
\]

### Step 5: Write the solutions
The solutions to the equation \( x^2 + 5x + 6 = 0 \) are:
\[
x = -2 \quad \text{and} \quad x = -3
\]

Thus, the values of \( x \) that satisfy the equation are \( x = -2 \) and \( x = -3 \).