Steps for Factoring Quadratic Equations

Steps for Factoring Quadratic Equations

1. Is the expression a binomail?

   • YES
     • Are the terms in the binomial perfect squares with subtraction between them? Ie: 16x^2 – 9
       • YES
         • Use the difference of squares method
           • Bracket both terms together twice with opposite signs. (16x^2 - 9) (16x^2 + 9). Calculate the square root of each term. The answer will be the same terms in brackets with Opposite signs. (4x – 3) (4x + 3)
         • NO
           • Factor out GCF to simplify expression
   • NO
     • Is the expression a trinomial?
       • YES
         • Factor out GCF if possible first
           • Is the expression a simple trinomial? form y= ax^2 + bx + c , a = 1
             • YES
               • Use sum & product method. Determine 2 numbers that multiply to get C and add to get B. Write the numbers from step 3 in binomials with the correct variables. Ie: x^2 – 4x – 12 -6 x 2 = -12 -6 + 2 = -4 =(x - 6) (x + 2)
               • NO
                 • Is the trinomial a perfect square? The first and last term must be perfect squares. The middle term must be double the product of the Roots of the other terms. Ie: 49x^2 + 84x + 36 Square root of 49x^2 = 7x Square root of 36= 6 = 2(7x) (6) =2(42x) =84x
                   • YES
                     • Use perfect square trinomial method
                       • Bracket the third and last term with the sign matching the middle term in the expanded equation. In factored form Both brackets must have the same terms. (7x + 6) (7x + 6) Calculate the square root of each number in the brackets. The answer will be the same terms in brackets squared. Ie: = (7x +6)^2
                     • NO
                       • Is the expression a complex trinomial? In the form y=ax^2 + bx + x , a is not equal to 1. Ie: 5x^2 + 22 + 8
                         • YES - Decide between Decomposition Method or Trial and Error/Inspection Method.
                           • Decomposition Method. 5x^2 + 22+ 8 Find two numbers that add to (sum) B and multiply (product) to “ac” - Example : AC = 5 x 8 = 40 B = 22 - ___ x ___ = 40 ____ + ____ = 22 20 x 2 = 40 20 + 2 = 22. Decompose (break down the middle term) Term will be written as 5x^2 + 20x + 2x + 8. Factor out the GCF in the first two terms and last two terms. 5x^2 + 20x = GCF is 5x 2x + 8 = GCF is 2. After GCF is factored out we have = 5x( 1x + 4) 2( 1x + 4). Use the sign on the third term to decide which operation to use and rewrite the factor This becomes: = 5x (x + 4)+ 2 (x +4). Factor out the binomial in the bracket = (5x + 2) (x + 4)
                           • Trial and Error/ Inspection Method - EXAMPLE 6x^2 + 27x + 12. Find 2 factors for the first number in the trinomial and Two factors for the third number. 6 = 2 & 3, 12 = 1 & 12. Multiply the numbers diagonally. 2 x 12 = 24 , 3x1 = 3. Add these 2 numbers together. 24+3=27 Compare the number with the coefficient of the middle Term. Continue this process until the number matches the Coefficient of the middle term. Use the first number listed as a factor for the first number in the trinomial and the first number listed as a factor for the third number in the trinomial and write the first binomial. (2x + 1) Now use the second number listed as a factor for the first number in the trinomial and the second number listed as a factor for the last number in the trinomial and write the second binomial. (3x + 12)Note: If the first term of the trinomial includes x it will be brought into the two binomials. (2x + 1) (3x + 12)
                             • Check your work by expanding out the binomial to see if the product of the two binomials is equal to the original trinomial. (2x + 1) (3x + 12) = 6x^2 + 27x + 12