Factoring and Solving Quadratic Equations

Quadratic Equations and Factoring Methods

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of degree 2, typically written in the form:

$ax^2 + bx + c = 0$

• Standard Form: All terms are on one side and ordered by descending powers of x.

• Quadratic Term: The term with $x^2$.

• Linear Term: The term with $x$.

• Constant Term: The term without $x$.

Example: Write $2x^2 - 2x - 6 = 0$ in standard form and identify $a$, $b$, and $c$:

• $a = 2$

• $b = -2$

• $c = -6$

Factoring Quadratic Equations

Factoring is a method used to solve quadratic equations by expressing the equation as a product of its factors and setting each factor equal to zero.

• To solve $ax^2 + bx + c = 0$, factor the left side and set each factor equal to zero.

• The solutions are called roots or zeros of the equation.

Factoring Process Flowchart

To choose a factoring method, follow these steps:

1. Factor out the Greatest Common Factor (GCF) first.

2. Count the number of terms:

   • 2 terms: Use special factoring formulas (difference of squares, sum/difference of cubes).

   • 3 terms: Check if it matches a factoring formula; if not, use the AC Method.

   • 4 terms: Use grouping.

Factoring Formula	Expression
Difference of Squares	$a^2 - b^2 = (a - b)(a + b)$
Sum of Cubes	$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
Difference of Cubes	$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Solving Quadratic Equations by Factoring

To solve a quadratic equation by factoring:

1. Write the equation in standard form: $ax^2 + bx + c = 0$.

2. Factor the quadratic expression.

3. Set each factor equal to zero and solve for $x$.

Example: Solve $x^2 - 5x + 6 = 0$ by factoring:

• Factor: $(x - 2)(x - 3) = 0$

• Set each factor to zero: $x - 2 = 0$ or $x - 3 = 0$

• Solutions: $x = 2$, $x = 3$

Factoring Guide: Steps

1. Write the equation in standard form.

2. Factor completely.

3. Set each factor equal to zero and solve for $x$.

4. Check your solutions in the original equation.

Practice Problems

• Solve $x^2 + 12x = 0$ by factoring.

• Solve $x^2 + 7x + 6 = 0$ by factoring.

Additional info:

• The AC Method is used for trinomials that do not fit special factoring formulas.

• Grouping is used for four-term polynomials.