Steps for Factoring Quadratic Expressions

Steps for Factoring Quadratic Expressions

1. Step by Step process

   • (Factoring trinomials): Find two numbers that multiply to 𝑎 𝑐 ac and add to 𝑏 b. Rewrite the middle term using those numbers. Factor by grouping.
   • (Difference of squares): Recognize the pattern 𝑎 2 − 𝑏 2 a 2 −b 2 . Apply the identity ( 𝑎 − 𝑏 ) ( 𝑎 + 𝑏 ) (a−b)(a+b).

2. Types of Quadratic Expressions

   • Standard form: 𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 ax 2 +bx+c
   • Difference of squares: 𝑎 2 − 𝑏 2 a 2 −b 2
   • Perfect square trinomials: ( 𝑎 𝑥 + 𝑏 ) 2 (ax+b) 2

3. Factoring Methods

   • Factoring by grouping
   • Trial and error (or guess and check) for trinomials
   • Using the quadratic formula: for complex cases.
   • Completing the square: a method if other techniques don’t apply.

4. Examples

   • For grouping: 𝑥 2 + 5 𝑥 + 6 = ( 𝑥 + 2 ) ( 𝑥 + 3 ) x 2 +5x+6=(x+2)(x+3)
   • For difference of squares: 𝑥 2 − 9 = ( 𝑥 − 3 ) ( 𝑥 + 3 ) x 2 −9=(x−3)(x+3)
   • For quadratic formula: 𝑥 2 + 4 𝑥 + 1 = − 4 ± 4 2 − 4 ( 1 ) ( 1 ) 2 ( 1 ) x 2 +4x+1= 2(1) −4± 4 2 −4(1)(1) ​ ​

5. Yes or no:

   • Is the quadratic a perfect square?
     • Yes → Use perfect square trinomial method.
     • No → Continue.
   • Is there a common factor across all terms?
     • Yes → Factor out the GCF.
     • No → Continue.
   • Is it a difference of squares?
     • Yes → Apply the difference of squares method.
     • No → Try factoring by grouping or trial and error.