Sections 5-4 & 5-5: Factoring Special Products and Solving Quadratic Equations

Welcome to the notes for our January 23rd, 2018 class! Today, we're focusing on Sections 5-4 and 5-5, which cover factoring special products and different techniques for solving quadratic equations. Let's break down the key concepts.

Section 5-4: Factoring Special Products

In this section, we learn how to recognize and factor special product patterns. Mastering these patterns will significantly speed up your factoring abilities.

Key Patterns:

  • Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$
  • Perfect Square Trinomials:
    • $a^2 + 2ab + b^2 = (a + b)^2$
    • $a^2 - 2ab + b^2 = (a - b)^2$

Example: Factor $x^2 - 9$

Solution: Recognizing this as a difference of squares, where $a = x$ and $b = 3$, we have:

$$x^2 - 9 = (x + 3)(x - 3)$$

Example: Factor $4x^2 + 12x + 9$

Solution: Recognizing this as a perfect square trinomial, where $a = 2x$ and $b = 3$, we have:

$$4x^2 + 12x + 9 = (2x + 3)^2$$

Practice identifying these patterns! The more you work with them, the easier they become.

Section 5-5: Solving Quadratic Equations by Factoring

Now, let's use our factoring skills to solve quadratic equations. A quadratic equation is an equation of the form:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a \neq 0$.

The Zero Product Property:

This is a fundamental property we use to solve quadratic equations by factoring. It states that if the product of two factors is zero, then at least one of the factors must be zero.

In other words, if $AB = 0$, then $A = 0$ or $B = 0$ (or both).

Steps to Solve by Factoring:

  1. Set the equation equal to zero.
  2. Factor the quadratic expression.
  3. Apply the Zero Product Property: Set each factor equal to zero.
  4. Solve each resulting equation.
  5. Check your solutions in the original equation.

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

  1. The equation is already set to zero.
  2. Factor the quadratic: $x^2 - 5x + 6 = (x - 2)(x - 3)$
  3. Apply the Zero Product Property:
    • $x - 2 = 0$ or $x - 3 = 0$
  4. Solve each equation:
    • $x = 2$ or $x = 3$

Therefore, the solutions are $x = 2$ and $x = 3$.

Important Considerations:

  • Always check for a greatest common factor (GCF) before attempting other factoring methods.
  • Not all quadratic equations can be factored easily. In those cases, we'll learn other methods like the quadratic formula later on.

Keep practicing, and you'll become a factoring pro! Remember to review the PowerPoint notes for more examples and explanations. Good luck with your studies!