# What is the new Transforming Method to solve quadratic equations?

Jun 30, 2015

The new Transforming Method to solve quadratic equations

#### Explanation:

1. Solving equation type ${x}^{2} + b x + c = 0$ with a = 1.
Solving is finding 2 numbers knowing sum (-b) and product (c). Compose factor pairs of c by observing the Rule of Signs. Then, find the pair whose sum is (-b).
Example 1: Solve x^2 - 31x - 102 = 0. Roots have different signs.
Compose factor pairs of c = -102--> (-2, 51)(-3, 34). This sum is 31 = -b. Then the 2 real roots are: (-3 and 34). No factoring, no solving binomials.
Example 2. Solve x^2 - 28x + 96 = 0. Both roots are positive.
Factor pairs of c = 96 -> (2, 48)(3, 32)(4, 24). This sum is 28 = -b. Then the 2 real roots are: 4 and 24.
Example 3. Solve x^2 + 39x + 108 = 0. Both roots are negative.
Factor pairs of c = 108 -> (-2, -54)(-3, -36). This sum is -39 = -b. Then the 2 real roots are: -3 and - 36.
We will see Case 2, solving standard type ax^2 + bx + c = 0 in the next explanation.
Jun 30, 2015
1. Solving standard type $y = a {x}^{2} + b x + c = 0$ (1)
Bring this case to Case 1 by transforming the equation (1) to->
$y ' = {x}^{2} + b x + a . c = 0$ (2)
Find the 2 real roots y1 and y2 of equation (2). The 2 real roots of original equation (1) are: $x 1 = \frac{y 1}{a} \mathmr{and} x 2 = \frac{y 2}{a}$.
Example 1. Solve $y = 8 {x}^{2} - 22 x - 13 = 0$ (1).
Transformed equation $y ' = {x}^{2} - 22 x - 104 = 0$ (2). Roots have different signs.
Factor pairs of ac = -104 --> (-2, 53)(-4, 26). This sum is 22 = -b. The
2 real roots of (2) are: y1 = -4 and y2 = 26. Back to original equation (1), the 2 real roots are: $x 1 = \frac{y 1}{a} = - \frac{4}{8} = - \frac{1}{2}$ and $x 2 = \frac{y 2}{a} = \frac{26}{8} = \frac{13}{4}$.
Example 2. Solve $y = 24 {x}^{2} + 59 x + 36 = 0$ (1)
Transformed equation: $y ' = {x}^{2} + 59 x + 864 = 0$ (2). Both roots are negative.
Factor pairs of ac = 864 --> ...(-18, -48)(-24, -36)(-27, -32). This sum is
-59 = -b. Then y1 = -27 and y2 = -32. Then, $x 1 = - \frac{27}{24} = - \frac{9}{8}$ and $x 2 = - \frac{32}{24} = - \frac{4}{3}$.