New non-factoring AC Method:
Case 1: Solving equation type x^2 + bx + c = 0. Solving means finding 2 numbers knowing their sum (-b) and their product (c).
New method proceeds by composing factor pairs of c, and by applying the Rule of Signs for real roots.
Example 1. Solve x^2 - 11x - 102 = 0. The 2 roots have different signs. Compose factor pairs of c = -102 with all first numbers being negative.
Proceed:
(-1, 102)(-2, 51)(-3, 34)(-6, 17).
The last sum is -6 + 17 = 11 = -b. Then the 2 real roots are -6 and 17. No factoring!
Case 2: Solving equation type ax^2 + bx + c = 0 (1). New AC method proceeds to bring this case back to Case 1.
Convert equation (1) to equation (2): x^2 + bx + a*c = 0 (2). Solve (2) like in Case 1. Compose factor pairs of a*c then find the 2 real roots y_1 and y_2 of Equation (2). Next step, divide y_1 and y_2 by the coefficient a to get the 2 real roots x_1 and x_2 of original equation (1).
Example:
Solve f(x) = 8x^2 - 22x - 13 = 0
(1) (a*c = 8*(-13) = -104)
Solution:
Converted equation f'(x) = x^2 - 22x - 104 = 0 (2). Roots have different signs. Compose factor pairs of a*c = -104.
Proceed:
(-1, 104)(-2, 52)(-4, 26)
This last sum is -4 + 26 = 22 = -b
The 2 real roots of (2) are:
y_1 = -4 and y_2 = 26
Then, the 2 real roots of original equation (1) are:
x_1 = (y1)/a = -4/8 = -1/2 and x_2 = (y2)/a = 26/8 = 13/4
No factoring!.
This new AC Method avoids the lengthy factoring by grouping.
