Consider the polynomial expression `ax^2 + bx + c.`
When the constant c is positive, the polynomial can be factored if there are 2 factors of ac that add to be the absolute value of b.
When the constant c is negative, the polynomial can be factored if there are 2 factors of ac that have a difference of the absolute value of b.
`6a^2 + 7a + 2`
Are there 2 factors of 2(6) or 12, whose sum is 7 (middle number)? This is 3 and 4. Rewrite by problem and factor by grouping.
`6a^2+3a + 4a + 2`
`3a(a+1) + 2(a+1)`
Are there 2 factors of 2(15) or 30 whose difference is 7 middle number)? The 2 numbers are 3 and 10
`2x^2 + 10x - 3x - 15`
`2x(x+5) -3(x + 5)`
Advantages of the new and improved Factoring AC Method.
With the application of the Rule of Signs for Real Roots into its solving approach, this new method helps:
1. To know in advance the signs (+ or -) of the 2 real roots for a better solving approach.
2. To immediately obtain the 2 real roots in case a = 1 (equation type x^2 + bx + c = 0). There is no need to factor by grouping and to solve binomials for x.
Example 1. Solve: x^2 - 11x - 102 = 0.
Solution. Roots have different signs. Compose factors of c = -102 with all first number being negative. Proceeding: (-1, 102),(-2, 51),(-3, 34),(-6, 17). This last sum is: 17 - 6 = 11 = -b. Consequently, the 2 real roots are: -6 and 17. No factoring and solving binomials.
3. To simplify the composition of factor pairs of the product a*c by reducing in half the number of permutations, or test cases.
Example 2. Solve: 16x^2 - 55x + 21 = 0.
Solution. Both roots are positive. Compose factor pairs of a*c = 336 with all positive numbers. Proceeding: (1, 336)(2, 168)(4, 82)(6, 56)(7, 48). This last sum is (7 + 48) = 55 = -b. Then, b1 = -7 and b2 = -48. Next, factor by grouping:
16x^2 - 48x - 7x + 21 = 16x(x - 3) - 7(x - 3) = (x -3)(16x - 7) = 0. Solving binomials: x = 3 and x =7/16.
The existing AC method. So far it is the most popular method to solve a quadratic equation in standard form a^2 +bx + c = 0. However, it can be improved by applying the Rule of Signs for Real Roots into its solving process.
Recall the Rule of Signs.
a. If a and c have different signs, roots ave different signs.
b. If a and c have same sign, roots have same sign.
- If a and b have same sign, both roots are negative.
- If a and b have different signs, both roots are positive.
The new and improved Factoring AC Method
This method proceeds the same as the existing AC method, but it simplify the process by applying the Rule of Signs in its procedures.
A. When a = 1 - Solving equation type x^2 + bx + c = 0.
This method composes factor pairs of ac = c, and in the same time applies the Rule of Signs following these TIPS:
TIP 1. When roots have different signs, compose factors of c with all first numbers of the pairs being negative.
TIP 2. When roots have same sign, compose factors of c with all positive numbers if both roots are positive, or with all negative numbers if both roots are negative.
Example 1. Solve x^2 + 31x + 108 = 0. Solution. Both roots are negative. Compose factors of c = 108 with all negatve numbers. Proceeding: (-1, -108)(-2, -54)(-3, -36)(-4, -27). This last sum is: -31 = -b. Then, the 2 real roots are:-4 and -27. There is no need for factoring by grouping and solving binomials.
B. Solving equation type ax^2 + bx + c = 0.
This new method proceeds the same as the existing AC Method, but it adds the Rule of Signs into its solving process. It also follows the same 2 TIPS above mentioned.
Example 2. Solve: 24x^2 + 59x + 36 = 0. Solution. Both roots are positive. Compose factor pairs of a*c = 24*36 = 864 with all negative numbers and start composing from the middle of the chain to save time. Proceeding:....(-18, -48)(-24, -36)(-27, -32). This last sum is -59 = -b. Then, b1 = 27 and b2 = 32. Next, factor by grouping:
24x^2 + 27x + 32x + 36 = 8x(3x + 4) = 9(3x + 4) = (3x + 4)(8x + 9)= 0. Solving binomials: x = -4/3, and x = -9/8.