"Solve the quadratic equation (x^2 + 5x + 6 = 0) and provide a step-by-step explanation."

To solve the quadratic equation \( x^2 + 5x + 6 = 0 \), we can use the **factoring method**. Here is a step-by-step explanation:

### Step 1: Write the equation in standard form

The equation is already in standard form, which is:

\[
x^2 + 5x + 6 = 0
\]

### Step 2: Identify the coefficients

Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation \( ax^2 + bx + c = 0 \):

- \( a = 1 \)
- \( b = 5 \)
- \( c = 6 \)

### Step 3: Factor the quadratic expression

To factor the quadratic expression \( x^2 + 5x + 6 \), we look for two numbers that multiply to \( c = 6 \) and add up to \( b = 5 \).

The two numbers that meet these criteria are 2 and 3, because:

- \( 2 \cdot 3 = 6 \)
- \( 2 + 3 = 5 \)

Using these numbers, we can factor the quadratic expression as:

\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]

### Step 4: Set each factor to zero

Set each factor equal to zero to solve for \( x \):

1. \( x + 2 = 0 \)
2. \( x + 3 = 0 \)

### Step 5: Solve each equation

Solve each equation for \( x \):

1. \( x + 2 = 0 \)

   \[
   x = -2
   \]

2. \( x + 3 = 0 \)

   \[
   x = -3
   \]

### Solution

The solutions to the quadratic equation \( x^2 + 5x + 6 = 0 \) are:

\[
x = -2 \quad \text{and} \quad x = -3
\]

### Verification

To verify the solutions, substitute \( x = -2 \) and \( x = -3 \) back into the original equation:

1. For \( x = -2 \):

   \[
   (-2)^2 + 5(-2) + 6 = 4 - 10 + 6 = 0
   \]

2. For \( x = -3 \):

   \[
   (-3)^2 + 5(-3) + 6 = 9 - 15 + 6 = 0
   \]

Both solutions satisfy the original equation.

Would you like further details or have any questions?

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Here are some additional related questions you might find useful:

1. What is the quadratic formula, and how is it used?
2. Can all quadratic equations be solved by factoring?
3. How does completing the square solve a quadratic equation?
4. What is the vertex form of a quadratic equation?
5. How do you determine the number of real solutions a quadratic equation has?
6. What are the applications of quadratic equations in real life?
7. How do you find the vertex of a quadratic function?
8. What is the discriminant, and how does it relate to quadratic equations?

**Tip:** When factoring quadratics, practice identifying pairs of numbers that multiply to the constant term and add to the linear coefficient to improve your factoring skills.

Learn how to solve the quadratic equation x^2 + 5x + 6 = 0 step-by-step with detailed explanations. This problem involves factoring quadratic expressions and verifying solutions. Suitable for high school students and anyone preparing for algebra tests.

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