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.

Example 1:

`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)`

`(3a+2)(a+1)`

Exampe 2:

`2x^2+7x-15`

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)`

`(2x-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.