4.3 Solving Quadratic Equations by Factoring

(x + 3)(x + 5) = + 5x +3x +15 (x + 3)(x + 5) Recall multiplying these binomials to get the standard form for the equation of a quadratic function: (x + 3)(x + 5) = + 5x +3x +15 The “reverse” of this process is called factoring. Writing a trinomial as a product of two binomials is called factoring. (x + 3)(x + 5)

Factor Since the lead coefficient is 1, we need two numbers that multiply to –28 and add to –12. Factors of -28 -1,28 1,-28 -2,14 2,-14 -4,7 4,-7 Sum of Factors 27 -27 12 -12 3 -3 Therefore:

Factor the expression: = (x-3)(x+7) Cannot be factored

Factoring a Trinomial when the lead coefficient is not 1. We need a combination of factors of 3 and 10 that will give a middle term of –17. Our approach will guess and check. Here are some possible factorizations: This is the factorization we seek.

Special Factoring Patterns you should remember: Pattern Name Pattern Example Difference of Two Squares Perfect Square Trinomial

Factor the quadratic expression:

A monomial is an expression that has only one term A monomial is an expression that has only one term. As a first step to factoring, you should check to see whether the terms have a common monomial factor. Factor:

You can use factoring to solve certain quadratic equation You can use factoring to solve certain quadratic equation. A quadratic equation in one variable can be written in the form where This is called the standard form of the equation: If this equation can be factored then we can use this zero product property. Zero Product Property Let A and B be real number or algebraic expressions. If AB = 0 the either A=0 or B=0

Solve: So, either (x+6)=0 x = -6 Or (x – 3)=0 x = 3 The solutions are –6 and 3. These solutions are also called zeros of the function Notice the zeros are the x-intercepts of the graph of the function.