Explain how factoring is used to solve quadratic equations. Demonstrate the process with an example.

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Factoring is a method used to solve quadratic equations by breaking down the equation into its factors, which are then set equal to zero to find the values of x that satisfy the equation. The process involves finding two numbers that multiply to give you the constant term of the quadratic equation and add up to give you the coefficient of the linear term.

Let's take the quadratic equation as an example: x^2 + 5x + 6 = 0

1. First, we need to find two numbers that multiply to give 6 and add up to give 5. In this case, the numbers are 2 and 3, as 2 * 3 = 6 and 2 + 3 = 5.

2. Re-write the middle term (5x) using these two numbers:
x^2 + 2x + 3x + 6 = 0

3. Group the terms:
(x^2 + 2x) + (3x + 6) = 0

4. Factor out the common factor from each group:
x(x + 2) + 3(x + 2) = 0

5. Notice that we now have a common factor, (x + 2), in both terms. We can factor it out:
(x + 2)(x + 3) = 0

6. Now, set each factor equal to zero:
x + 2 = 0 or x + 3 = 0

7. Solve for x:
x = -2 or x = -3

Therefore, the solutions to the quadratic equation x^2 + 5x + 6 = 0 are x = -2 and x = -3.

By factoring the quadratic equation, we were able to find the values of x that satisfy the equation and solve it.