• Solving quadratic equations by the new Transforming Method

Solving quadratic equations by the new Transforming Method.

  • By Nghi H Nguyen, co-author of the Diagonal Sum Method.
  • GENERALITIES.

This new method is may be the simplest and fastest method to solve a quadratic equation in standard form that can be factored. Its strong points are: simple, fast, systematic, no guessing, no factoring by grouping, and no solving binomials. It proceeds based on 3 features:

  • - The Rule of Signs for Real Roots of a quadratic equation that show the signs (+ or -) of the real roots to select a better solving approach.
  • - The Diagonal Sum Method (Google Search) to solve quadratic equations in simplified form, when a = 1-
  • - The transformation of a quadratic equation in standard form ax^2 + bx + c = 0 into a simplified equation that has the form x^2 + bx + a*c = 0.
  • RECALL THE RULE OF SIGNS.
  • A. When a and c have different signs, roots have different signs.
  • B. When a and c have same sign, roots have same sign,
 a. If a and b have different signs, both roots are positive.
 b. If a and b have same sign, both roots are negative.
  • THE DIAGONAL SUM METHOD TO SOLVE EQUATION TYPE x^2 + bx + c = 0.

When a = 1, solving results in solving 2 numbers knowing the sum (-b) and the product (c). This method can immediately obtain the 2 real roots without factoring and solving binomials. This method proceeds by composing the factor pairs of (c), following these 2 Tips.

  • - TIP 1. If roots have different signs, compose factor pairs of (c) with all first numbers being negative.
  • Example 1. Solve: x^2 - 11x - 102 = 0. Roots have different signs. Compose factor pair of c = -102. Proceeding: (-1, 102)(-2, 51)(-3, 34)(-6, 17). This last sum is; -6 + 17 = 11 = -b. Then, the 2 real roots are -6 and 17. no factoring and solving binomials.
  • - TIP 2. If roots have same sign, compose factor pairs of (c) :
  - with all positive numbers, when both roots are positive
  - with all negative numbers when both roots are negative.
  • Example 2. Solve: x^2 - 21x + 108 = 0. Both roots are positive. Compose factor pairs of c = 108 with all positive numbers. When c is a large number, start composing from the middle of the chain to save time. Proceeding:.....(4, 27)(6, 18)(9, 12). This last sum is 21 = -b. Then, the 2 real roots are 9 and 12.
  • Example 3. Solve x^2 + 22x + 117 = 0. Both roots are negative. Compose factor pairs of c = 117 with all negative numbers. Proceeding: (-3, -39)(-9, -13). This last sum is; -22 = -b. The 2 real roots are: -9 and -13.

THE NEW TRANSFORMING METHOD.

  • It proceeds through 3 steps..
  • 1. STEP 1'. Transform the quadratic equation in standard form ax^2 + bx + c = 0 (1) into a simplified equation with a = 1. This transformed equation has the form: x^2 + bx + a*c = 0.
  • 2. STEP2. Solve the transformed equation (2) by the Diagonal Sum Method that immediately obtains the 2 real roots, without factoring and solving binomials. Suppose the 2 obtained real roots of (2) are: y1, and y2.
  • 3. STEP 3. Divide both y1 and y2 by the constant a to get the 2 real roots x1, and x2 of the original equation (1).
  • Example 4. Original equation to solve: 6x^2 - 11x - 35 = 0. (1). Transformed equation: x^2 - 11x - 210 = 0. (2). Solve (2) by the Diagonal Sum Method. Compose factor pairs of a*c = -210. Proceeding: .....(-5, 42)(-7, 30)(-10, 21). This last sum is: 21 - 10 = 11 = -b. Then, the 2 real roots of (2) are: y1 = -10, and y2 = 21. Next, find the 2 real roots of the original equation: x1 = y1/6 = -10/6 = -5/3, and x2 = y2/6 = 21/6 = 7/2.
  • Example 5. Original equation to solve: 12x^2 + 29x + 15 = 0 (1). Transformed equation: x^2 + 29x + 180 = 0 (2)

Both roots are negative. Compose factor pairs of a*c = 180. Start composing from the middle of the factor chain. Proceeding: .....(-5, -36)(-6, -30)(-9, -20). This last sum is -29 = -b. The 2 real roots are: y1 = -9, and y2 = -20. Next, find the 2 real roots of the original equation (1): x1 = y1/12 = -9/12 = -3/4, and x2 = y2/12 = -20/12 = -5/3.

  • Example 6. Solve original equation: 16x^2 - 55x + 21 = 0. Transformed equation: x^2 - 55x + 336 = 0. Both roots are positive. Compose factor pairs of a*c = 336 with all positive numbers. Start from the middle of the chain. Proceeding:..... (4, 82)(6, 56)(7, 48). This sum is: 55 = -b. Then, the 2 real roots of the equation (2) are: y1 = 7 and y2 = 48. Next, find the 2 real roots of the original equation: x1 = 7/16, and x2 = 48/16 = 3.
  • References:
  • - The Diagonal Sum Method (Yahoo or Google Search)
  • - Transformation of original quadratic equation in standard form into simplified form, with a = 1. (Google Search)
 - Factoring Trinomials - Simplified Method at 2000clicks.com/mathhelp.
 - AC method for factoring Trinomials - Regent Exam Prep Center at: www.regentprep.org/regent/math
  • - The Bluma Method (Yahoo or Google Search).
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