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.
