# What is the new non-factoring AC Method to solve quadratic equations?

Apr 11, 2015

New non-factoring AC Method:

Case 1: Solving equation type ${x}^{2} + b x + c = 0$. 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 ${x}^{2} - 11 x - 102 = 0$. The 2 roots have different signs. Compose factor pairs of $c = - 102$ with all first numbers being negative.

Proceed:

$\left(- 1 , 102\right) \left(- 2 , 51\right) \left(- 3 , 34\right) \left(- 6 , 17\right)$.

The last sum is $- 6 + 17 = 11 = - b$. Then the 2 real roots are $- 6$ and $17$. No factoring!

Case 2: Solving equation type $a {x}^{2} + b x + c = 0 \left(1\right)$. New AC method proceeds to bring this case back to Case 1.

Convert equation (1) to equation (2): ${x}^{2} + b x + a \cdot c = 0 \left(2\right)$. Solve (2) like in Case 1. Compose factor pairs of $a \cdot c$ 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 $f \left(x\right) = 8 {x}^{2} - 22 x - 13 = 0$

(1) $\left(a \cdot c = 8 \cdot \left(- 13\right) = - 104\right)$

Solution:

Converted equation $f ' \left(x\right) = {x}^{2} - 22 x - 104 = 0 \left(2\right)$. Roots have different signs. Compose factor pairs of $a \cdot c = - 104$.

Proceed:

$\left(- 1 , 104\right) \left(- 2 , 52\right) \left(- 4 , 26\right)$

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:

${x}_{1} = \frac{y 1}{a} = - \frac{4}{8} = - \frac{1}{2}$ and ${x}_{2} = \frac{y 2}{a} = \frac{26}{8} = \frac{13}{4}$

No factoring!.

This new AC Method avoids the lengthy factoring by grouping.