How to Solve Quadratic Equations with the "Transforming Method"

Step-by-Step Guide

1. 1

   Step 1: Recall the Rule of Signs.

   If a and c have different signs, roots have different signs If a and c have same sign, roots have same sign.
   
   If a and b have different signs, both roots are positive.
   
   If a and b have same sign, both roots are negative.
2. 2

   Step 2: Transform the equation in standard form ax^2 + bx + c = 0 (1) into a new equation

   , Solving results in finding 2 numbers knowing the sum (-b) and the product (a*c).
   
   Compose factor pairs of a*c following these 2 Tips below.
   
   Find the pair that equals to (-b), or b.
   
   If you can't find this pair, this means the equation can't be factored, and you should probably solve it by the Quadratic Formula.
   
   If roots have different signs (a and c different signs), compose factor pairs of a*c with all first numbers being negative.
   
   If roots have same sign (a and c same sign), compose factor pairs of a*c: with all negative numbers when both roots are negative. with all positive numbers when both roots are positive.
   
   Example
   1.
   
   Solve: x^2
   - 11x
   - 102 =
   0.
   
   Roots have different signs.
   
   Compose factor pairs of c =
   -102 with all first numbers being negative.
   
   Proceeding: (-1, 102)(-2, 51)(-3, 34)(--6, 17).
   
   This last sum is: 17
   - 6 = 11 =
   -b.
   
   Then, the 2 real roots are:
   -6 and
   17.
   
   No factoring and solving binomials.
   
   Example
   2.
   
   Solve: x^2 + 39x + 108 =
   0.
   
   Both roots are negative.
   
   Compose factor pairs of c = 108 with all negative numbers.
   
   Proceeding: (-1,
   -108)(-2,
   -54)(-3,
   -36).
   
   This last sum is
   -39 =
   -b.
   
   Then, the 2 real roots are:
   -3, and
   -36. " Example 3".
   
   Solve: x^2
   - 23x + 102 =
   0.
   
   Both roots are positive.
   
   Compose factor pairs of c = 102 with all positive numbers.
   
   Proceeding: (1, 102)(2, 51)(3, 34)(6, 17).
   
   This last sum is: 17 + 6 = 23 =
   -b.
   
   The 2 real roots are: 6 and
   17. ,, Examples of solving by the new "Transforming Method" Example
   3.
   
   Original equation to solve: 6x^2
   - 19x
   - 11 =
   0. (1).
   
   First, solve the transformed equation: x^2
   - 19x
   - 66 =
   0.(2).
   
   Roots have different signs.
   
   Compose factor pairs of a*c =
   -66.
   
   Proceeding: (-1, 66)(-2, 33)(-3, 22).
   
   This last sum is 22
   - 3 = 19 =
   -b.
   
   Then, the 2 real roots of (2) is: y1 =
   -3, and y2 =
   22.
   
   Next, divide both y1, and y2 by a =
   6.
   
   The 2 real roots of the original equation (1) are: x1 = y1/6 =
   -3/6 =
   -1/2, and x2 = y2/6 = 22/6 = 11/3.
   
   Example
   4.
   
   Original equation to solve: 6x^2
   - 11x
   - 35 = 0 (1). , Roots have different signs.
   
   To save time, compose factor pairs from the middle of the factor chain.
   
   Proceeding: .....(-5, 42)(-7, 30)(-10, 21).
   
   This last sum is: 21
   - 10 = 11 =
   -b.
   
   Then, y1 =
   -10, and y2 =
   21.
   
   Next, find the 2 real roots of the original equation (1): x1 = y1/6 =
   -10/6 =
   -5/3, and x2 = 21/6 = 7/2..
   
   Example
   5.
   
   Original equation: 12x^2 + 29x + 15 =
   0. (1).
   
   Solve transformed equation: x^2 + 29x + 180 = 0 (2).
   
   Both roots are negative.
   
   Start composing a*c = 180 from the middle of the factor chain.
   
   Proceeding:..... (-5,
   -36)(-6,
   -30)(-9,
   -20).
   
   This last sum is:
   -29 =
   -b.
   
   The 2 real roots of (2) are: y1 =
   -9, and y2 =
   -20.
   
   Next, find the 2 real roots of (1): x1 =
   -9/12 =
   -3/4, and x2 =
   -20/12 =
   -5/3.
3. 3

   Step 3: with a = 1

4. 4

   Step 4: and the constant C = a*c. The new equation has the form: x^2 + bx + a*c = 0

5. 5

   Step 5: Solve the transformed equation (2) by the Diagonal Sum Method that can immediately obtain the 2 real roots.

6. 6

   Step 6: Assume that the 2 real roots of the simplified equation (2) are: y1

7. 7

   Step 7: and y2.

8. 8

   Step 8: Divide both real roots y1 and y2 by the coefficient a to get the 2 real roots x1

9. 9

   Step 9: and x2 of the original equation (1).

10. 10

    Step 10: Solve the transformed equation: x^2 - 11x - 210 = 0 (2).

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