Enhanced Math Problem: Solving a Quadratic Equation

Solve the quadratic equation $x^2 + 5x + 6 = 0$ and show all the steps involved.

The solution should:

Clearly present each step in the problem-solving process using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 5$, and $c = 6$.

Provide explanations for each step, using mathematical terminology and symbols correctly, such as describing the use of the quadratic formula and its components.

Arrive at the correct answer and, if applicable, simplify the solution by factoring or using the quadratic formula.

Specify the method or approach to be used: the quadratic formula will be employed to solve the equation.

Request a visual representation of the solution: a graph of the quadratic function $f(x) = x^2 + 5x + 6$ can be used to visualize the roots of the equation, which correspond to the solutions of the equation $x^2 + 5x + 6 = 0$.

To solve the equation, we will follow these steps:

1. Write down the given equation: $x^2 + 5x + 6 = 0$.

2. Identify the coefficients $a$, $b$, and $c$: $a = 1$, $b = 5$, and $c = 6$.

3. Apply the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

4. Substitute the values of $a$, $b$, and $c$ into the quadratic formula: $x = \frac{-5 \pm \sqrt{5^2 - 4(1)(6)}}{2(1)}$.

5. Simplify the expression under the square root: $x = \frac{-5 \pm \sqrt{25 - 24}}{2}$.

6. Further simplify the expression: $x = \frac{-5 \pm \sqrt{1}}{2}$.

7. Solve for both values of $x$: $x_1 = \frac{-5 + 1}{2} = -2$ and $x_2 = \frac{-5 - 1}{2} = -3$.

8. Write the final solutions: $x = -2$ or $x = -3$.

The solutions to the equation $x^2 + 5x + 6 = 0$ are $x = -2$ and $x = -3$. A graph of the function $f(x) = x^2 + 5x + 6$ will show that the roots of the equation are $x = -2$ and $x = -3$, confirming the solutions obtained using the quadratic formula.