Steps for Factoring Quadratics

Steps for Factoring Quadratics

1. Does your equation look like…

   • (x+6)(x-6) a binomial with opposite signs
     • This can be solved using the difference of squares method!
       • (x)(x)+(x)(-6)+(6)(x)+(6)(-6) First, you expand the expression. You do this by multiplying everything in the first brackets with everything in the second.
         • x²-6x+6x-36 Then you combine like terms, in a difference of squares combine the 2 terms in the middle as they are opposites of each other and thus, cancel each other out.
           • =x²-36
     • or (x ²-9)
     • difference of squares always include 2 of the same terms with opposite signs
   • (x+6)(x+6) a binomial with the same signs
     • This can be solved using the perfect square trinomial method
       • (x)(x)+(x)(6)+(6)(x)+(6)(6) First, you expand the expression. You do this by multiplying everything in the first brackets with everything in the second.
         • =x²6x+6x+36 Then you combine like terms, in a perfect square trinomial, the 2 terms in the middle always keep their original sign as they are the same.
           • The middle terms can be added as they are like terms: x²+12x+36
     • perfect square trinomials always include 2 of the same terms with the same signs
   • (x ²+7x+12) a simple trinomial
     • This can be solved using the sum and product method!
       • (4x3=12) last term (4+3=7) middle term Find 2 numbers that multiply together to create the last term and add together to create the middle term.
         • =x²+7x+12 =(x+4)(x+3)
           • what if my equation looks like x²-9
             • You could use the difference of squares method because it satisfies the 2 terms and the last term creates perfect squared terms (-3x and 3x)
               • factor x²-9 =(x-3)(x+3)
     • simple trinomials follow the formula ax²+bx+c, a=1
   • (3x ²+5x+2) a complex trinomial
     • This can be solved using the decomposition method
       • 3x²+5x+2 Find 2 terms that add to the middle term(b) and when multiplied equal the sum of the first term and last term multiplied (a x c)
         • what if my equation looks like: 4x ²-12x+9
           • you can use the perfect square trinomials method to solve this as well! (applies im cases where the first term and last term and perfect squares)
             • 4x²-12x+9 =4x²+6x-6x+9 =2x(2x-3)-3(2x-3) =(2x-3)(2x-3) =(2x-3)²
           • if solved using decomposition a²+2ab+b²=(a-b)² =2x²(-2)(2x)(3)+3² =(2x-3)²
         • 3+2=5 (b) 3x2=6 (a x c) (new terms are 3 and 2)
           • 3x²+3x+2x+2 You break down the middle term using the 2 new terms.
             • Then you factor out the GCF in the first 2 terms and then the GCF of the remaining 2 terms.
               • 1st GCF is 3x, 2nd GCF is 2 = 3x(x+1)+2(x+1)
                 • Factor out (x+1) = (x+1)(3x+2) Then factor out the binomial in the brackets.
     • complex trinomials follow the formula ax²+bx+c, a does not equal 1