How do you solve quadratic equations by the new Transforming Method?

Answer 1

Generalties. This new method is may be the simplest and fastest method to solve quadratic equations in standard form that can be factored. Its solving procees bases on 3 features:1

a. The Rule of Signs for real roots of a quadratic equation.

b. The Diagonal Sum method (Google or Yahoo Search) to solve equation type x^2 + bx + c = 0, when a =1.

c. The transformation of a quadratic equation type ax^2 + bx + c = 0 into the simplified type x^2 + bx + c = 0, with a = 1.

The Rule of Signs.

a. If a and c have different signs, roots have different signs.

b. If a and c have same sign, roots have same sign;

- When a and b have diffrent signs, both roots are positive.

- When 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 finding 2 numbers knowing the sum (-b) and the product (c). This method can immediately obtain the 2 real roots without factoring by grouping and solving binomials. It proceeds composing factor pairs of c, following these 3 Tips.

TIP 1. When 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 pairs of c = -102 with all first numbers being negative. 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 no solving binomials!

TIP 2. When both roots are positive, compose factor pairs of c with all positive numbers.

Example 2. Solve: x^2 - 31x + 108 = 0. Both roots are positive. Proceeding:(1, 108)(2, 54)(3, 36)(4, 27). This last sum is 4 + 27 = 31 = -b. Then, the 2 real toots are: 4 and 27.

Tip 3. When both roots are negative, compose factor paits with all negative numbers.

Example 3. Solve: x^2 + 62x + 336 = 0. Both roots are negative. Compose factor pairs of c = 336 with all negative numbers. Proceeding: (-1, -336)(-2, -168)(-4, -82)(-6, -56). This last sum is: -6 - 56 = -62 = -b. Then the 2 real roots are: -6 and -56.

The new "Transforming Method" to solve quadratic equations.

It proceeds through 3 Steps.

STEP 1. Transform the original equation in standard form ax^2 + bx + c = 0. (1) into a simplified equation, with a = 1 and C = a*c. The transformed equation has the form: x^2 + bx + a*c = 0. (2).

STEP 2. Solve the transformed equation (2) by the Diagonal Sum Method that immediately obtains the 2 real roots. Suppose they are: y1 , and y2.

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): x1 = y1/a, and x2 = y2/a.

Example 4. Original equation to solve: 8x^2 - 22x - 13 = 0. (1). Transformed equation: x^2 - 22x - 104 = 0.(2). Roots have different signs. Compose factor pairs of a*c = -104. Proceeding:(-1, 104)(-2, 52)(-4, 26). This last sum is 26 - 4 = 22 = -b. The 2 real roots of the transformed equation are: y1 = -4 and y2 = 26. Next, find the 2 real roots of the original equation (1): x = y1/8 = -4/8 = -1/2, and x2 = y2/8 = 26/8 = 13/4.

Example 5. Solve: 15x^2 - 53x + 16 = 0. Both roots are positive. Compose factor pairs of a*c = 240 from the middle of the chain to save time. Proceeding:....(3, 80)(4, 60)(5, 48). This last sum is 5 + 48 = 53 = -b. Then, y1 = 5 and y2 = 48. Next, find x1 = y1/15 = 5/15 = 1/3, and x2 = y2/15 = 48/15 = 16/5.

Conclusion. The strong points of this new method are: simple, fast, systematic, no guessing, no factoring by grouping, and no solving binomials.