How can you solve a quadratic equation in one variable using the factoring method?

The factoring method solves quadratic equations by rewriting them as a product of two binomials, then setting each binomial equal to zero to find the solutions (roots). Here's a detailed explanation:

Steps to Solve a Quadratic Equation by Factoring

1. Write the equation in standard form: The quadratic equation must be in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.

2. Factor the quadratic expression: Find two binomials that multiply to give the quadratic expression. This involves finding two numbers that:

   • Multiply to give ac (the product of the coefficient of x² and the constant term).
   • Add up to b (the coefficient of the x term).

   There are several factoring techniques, including:

   • Simple Trinomials (a=1): If a = 1, find two numbers that multiply to c and add up to b. For example, to factor x² + 5x + 6, you need numbers that multiply to 6 and add to 5. These numbers are 2 and 3, so the factored form is (x + 2)(x + 3).

   • Complex Trinomials (a≠1): If a ≠ 1, this can be more involved. The 'ac' method is often used. For example, to factor 2x² + 7x + 3, ac = 23 = 6. You need numbers that multiply to 6 and add to 7. These numbers are 1 and 6. Rewrite the middle term using these numbers: 2x² + x + 6x + 3. Then, factor by grouping: x(2x + 1) + 3(2x + 1) = (2x + 1)(x* + 3).

   • Difference of Squares: If the quadratic is in the form a² - b², it factors as (a + b) (a - b). For example, x² - 9 factors as (x + 3)(x - 3).

3. Set each factor equal to zero: Once you have the equation in factored form (e.g., (x + 2)(x + 3) = 0), set each factor equal to zero:

   • x + 2 = 0
   • x + 3 = 0

4. Solve for x: Solve each equation to find the values of x.

   • x = -2
   • x = -3

   These values of x are the solutions (or roots) of the quadratic equation.

Example

Solve the quadratic equation x² - 5x + 6 = 0 by factoring.

1. Standard Form: The equation is already in standard form.

2. Factor: Find two numbers that multiply to 6 and add to -5. These numbers are -2 and -3. Therefore, the factored form is (x - 2)(x - 3) = 0.

3. Set factors to zero:

   • x - 2 = 0
   • x - 3 = 0

4. Solve for x:

   • x = 2
   • x = 3

Therefore, the solutions to the quadratic equation x² - 5x + 6 = 0 are x = 2 and x = 3.

When Factoring Works Best

The factoring method is most efficient when the quadratic equation has integer roots and the coefficients are relatively small, making it easier to find the factors. If factoring is difficult or impossible with integers, other methods like the quadratic formula or completing the square are more appropriate.