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There is a new solving method, recently introduced, that is much faster and simpler than the existing factoring AC Method. See articles titled: "Solving quadratic equations by the new Transforming Method" on Yahoo or Google Search. This method can immediately obtain the 2 real roots without factoring by grouping and solving the 2 binomials. It uses 3 features in its solving process:
The AC Method (YouTube), the factoring method by grouping, has been the most popular systematic method to solve quadratic equations in standard form ax^2 + bx + c = 0. However, it can be considerably improved if the Rule of Signs for Real Roots of a quadratic equation be added to its solving process. There is a "new and improved factoring AC Method" recently introduced on Google or Yahoo Search, that presents many advantages:
Solving quadratic equations by the new Transforming Method,
This new method works through 3 steps.
STEP 1. Transform the given quadratic equation ax^2 + bx + c = 0 (1) into the simplified form x^2 + bx + a*c = 0 (2), with a = 1 , and C = a*c.
STEP 2. Solve the transformed equation (2) by the Diagonal Sum Method that immediately obtains the 2 real roots y1 and y2 of the equation (2).
STEP 3. Divide both y1 and y2 by the coefficient a to get the 2 real roots x1 and x2 of the original equation: x1 = y1/a, and x2 = y2/a.
Example from the video. Solve: 25x^2 - 20x + 4 = 0 (1).
Step1. Transform the equation (1) to the form: x^2 - 20x + 100 = 0 (2).
Step 2. Solve the transformed equation by the Diagonal Sum Method (Google or Yahoo Search) when a = 1. Both roots are positive (Rule of Signs). Compose factor pairs of a*c = 100 with all positive numbers. Proceeding: (1, 100)(2, 50)(4, 25) (5, 20)(10, 10). This sum is 10 + 10 = 20 = -b. The double real root of the equation (2) is y1 = y2 = 10.
Back to the original equation (1), the double real root is: x1 = x2 = 10/25 = 2/5.
Example 2. Solve: 12x^2 + 5x - 72 = 0 (1).
Step 1. Transformed equation: x^2 + 5x - 864 = 0 (2)
Step 2. Solve equation (2) by the Diagonal Sum Method. Roots have different signs (Rule of Signs). Compose factor pairs of ac = -864 with all first numbers being negative. Start composing from the middle of the chain to save time. Proceeding: ....(-18, 48)(-24, 36)(-27, 32). This last sum is -27 + 32 = 5 = b. Then, the 2 real roots of the equation (2) are: y1 = 27 and y2 = -32 (Rule of the Diagonal Sum).
Step 3. Back to the original equation (1), the 2 real roots are: x1 = y1/ a = 27/12 = 9/4 and x2 = y2/12 = -32/12 = -8/3.
CONCLUSION. The strong points of the new Transforming Method are:
simple, fast, no guessing, systematic, no factoring by grouping and no solving binomials.
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