Calculus

12 + 3x = 4

12 - 12 + 3x = 4 - 12

3x = -8

3x/3 = -8/3

x = -8/3

x = -(2 2/3)

This new Diagonal Sum Method is recently presented in book titled:"New methods for solving quadratic equations and inequalities" (Amazon e-book 2010)

It quickly and directly gives the 2 real roots of the equation without factoring. The innovative concept of the method is direct finding the 2 real roots, in the form of 2 fractions, knowing their Sum (-b/a) and their Product (c/a). It is fast and convenient and is applicable whenever a given quadratic equation can be factored. It is a trial-and-error method that can replace the existing trial-and-error factoring method since it is faster and has fewer permutations. Here is how this new method works:

A quadratic equations has as standard for: ax^2 + bx + c = 0, with a not zero.

Recall the Rule of Signs for real roots.

1. If a and c have opposite signs, the 2 real roots have opposite signs.

Example: The equation x^2 + 5x - 7 = 0 has 2 real roots with opposite signs

2. If a and c have same sign, both real roots have same sign.

a. If a and b have opposite signs, both roots are positive.

Example. The equation x^2 - 7x + 5 = 0 has 2 real roots both positive

b. If a and b have same sign, both roots are negative.

Example. The equation x^2 + 9x + 8 = 0 has 2 real roots both negative

The Diagonal Sum of a root pair.

Given a pair of real roots: (c1/a1) + (c2/a2). Their product is (c1,c2/a1.a2) = c/a.

This means the 2 numerators of a real root pair constitute a factor pair of c. The 2 denominators constitute a factor pair of a.

Their sum is: (c1/a1) + (c2/a2) = (c1a2 + c2a1)/a1a2 = -b/a.

The sum (c1a2 + c2a1) is called the diagonal sum of a probable root pair. It must be equal to -b in order to make the pair a real root pair.

The Rule of the Diagonal Sum.

The diagonal sum of a real root pair must be equal to -b. If it equals b, the answers are opposite. If a is negative, the Rule is reversal in signs.

How does the Diagonal Sum Method proceed?

It directly selects the probable root pairs from the (c/a) setup. The numerator of the setup contains all factor pairs of c. The denominator contains all factor pairs of a. Then, the new method uses mental math to calculate the diagonal sums of these probable root pairs to find the one that equals to -b.

To know in details on how to proceed, please read the article titled "How to solve quadratic equation by the new Diagonal Sum Method" on this Wiki Answers website.

NOTE. The new Diagonal Sum Method may be called "The c/a Method" since it proceeds solving, based on the c/a setup. It may be also called the shortcut of the factoring method, since it saves the time used to solve the 2 binomials in x.

Examples in solving by the new Diagonal Sum Method.

Solving quadratic equations is simple or complicated depending on the values of the constants a and c.

Case 1. When a = 1 - Solving equation type: x^2 + bx + c = 0. In this case, solving is simple and doesn't need factoring. Solving means finding 2 numbers knowing their sum (-b) and their product (c).

Example 1. Solve x^2 - 11x - 102 = 0.

Solution. The 2 real roots have opposite signs. Write factor pairs of c = -102.

They are: (-1, 102),(-2, 51),(-3, 34),(-6, 17).... Stop. This sum is 17 - 6 = 11 = -b. The 2 real roots are -6 and 17.

Note. By convention, when the roots have opposite signs, always put the negative sign in front of the first root. Exp: (-1, 102),(-2, 51).

Example 2. Solve -x^2 + 35x - 96 = 0.

Solution. Both real roots are positive. The constant a is negative. Write factor pairs of ac = 96. They are: (1, 96), (2, 48),(3, 32)... Stop. This sum is 32 + 3 = 35 = b. According to the Diagonal Sum Rule, when a is negative, the 2 real roots are 3 and 32. No need for solving the 2 binomials!

Case 2. When a and c are prime numbers. The c/a setup has the simplest form.

Example 3. Solve 7x^22 - 76x - 11 = 0.

Solution. Roots have opposite signs. Both a and c are prime numbers. The c/a setup: (-1, 11)/(1, 7). There is unique probable root pair: (-1/7 , 11/1), since -1 is not a real root of the given equation. The diagonal sum is: 77 - 1 = 76 = -b. The 2 real roots are -1/7 and 11.

Case 3. When a and c are small numbers and may have themselves 1 or 2 factors. In this care the c/a setup may have 1 or 2 factor pairs in its numerator or denominator. We can eliminate the factor pairs that do not fit to get to the c/a simplest form.

Example 4. Solve: 8x^2 - 22x - 13 = 0.

Solution. Roots have opposite signs. The c/a setup: (-1, 13)/(-1, 2)(-2, 4). We can eliminate the pair (-1, 2) since combined with any other pair, they will give an odd number diagonal sum (while b is even0. The remainder c/a gives as 2 probable root pairs: (-1/2, 13/4) and (-1/4 , 13/2). The second diagonal sum is -4 + 26 = 22 = -b. The 2 real roots are -1/2 and 13/4.

Case 4. When a and c are large numbers and may have themselves many factor pairs. This case is considered complicated since the c/a setup contains many factor pairs in both of its numerator and denominator. The Diagonal Sum Method can then transform a multiple steps solving process into a simplified one by eliminating the factor pairs that don't fit.

Example 5. Solve: 12x^2 + 5x - 72 = 0.

Solution. Roots have opposite signs. Write the c/a setup:

Numerator: (-1, 72)(-2, 36)(-3, 24)(-4, 18)(-6, 12)(-8, 9)

Denominator: (1, 12)(2, 6) (3, 4)

Look to eliminate the pairs that don't fit. First, eliminate the pairs (-2, 36)(-4, 18)(-6, 12) from the numerator and the pair (2, 6) from the denominator because they give even-diagonal sums (while b = 5 is odd). Next, eliminate the pairs (-1, 72)(-3, 24)/(1, 12) because they give large diagonal sums (while b = 5 is small number).

The remainder c/a is (-8, 9)/(3/4) that gives 2 probable root pairs: (-8/3 , 9/4) and (-8/4 , 9/3). The first diagonal sum is 27 - 32 = -5 = -b. The 2 real roots are -8/3 and 9/4.

Conclusion. The Diagonal Sum Method directly finds the probable real root pairs by considering the c/a set up. When the c/a setup is complicated, this method transforms the setup into its simplest form by doing some eliminations. Then, by applying a simple formula, students can easily find the diagonal sum that fits.