How To: Solve quadratic equations by factoring using grouping

Solve quadratic equations by factoring using grouping

Learn how to take a quadratic equation in standard form and change the trinomial into a binomial product. Using the zero product rule, the zeros of the graph can then be found. The grouping method is effective when the leading coefficient (the number factor of the squared term) is some number other than 1 (one). From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just finish your homework or study for that next big test).

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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:

  1. The Rule of Signs for Real Roots.
  2. The Diagonal Sum Method to solve quadratic equation type x^2 + bx + c = 0, when a = 1
  3. The transformation of equation.

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:

  1. It shows the signs (+ or -) of the 2 real roots to select a better solving approach.
  2. When a = 1, it immediately obtains the 2real roots without factoring by grouping and solving the binomials.
  3. In general cases, a is not 1, it offers the logical and effective way to find the 2 numbers b1 and b2 for the factoring by grouping.

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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