How do you solve quadratic equations by the Diagonal Sum Method?

Answer 1

The standard form of a quadratic equation in one variable is ax^2 + bx + c = 0.

Beyond the 4 existing common solving methods, there is a new method, called the Diagonal Sum Method, presented in book titled:" New methods for solving quadratic equations and inequalities" (Amazon e-books 2010), that directly obtains the 2 real roots, in the form of 2 fractions, without factoring. It uses a Rule of signs for real roots and a Rule for the Diagonal Sum.

Example of using the Rule of signs for real roots.

The equation 15x^2 - 46x - 17 = has 2 real roots that have opposite signs.

The equation 21x^2 + 34x + 8 = 0 has 2 real roots, both negative.

The equation 21x^2 - 68x + 15 = 0 has 2 real roots, both positive.

The diagonal sum of a pair of 2 real roots.

Given a pair of two real roots: (c1/a1) and (c2/a1).

Their product is c1.c2/a1.a2 = c/a.

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

The sum (c1a2 + c2a1) is called the diagonal sum. The diagonal sum of the 2 real roots should be equal to -b.

Rule for the Diagonal Sum.

The diagonal sum of a true real root pair must be equal to (-b). If it is equal to b, the answers are opposite in sign. If a is negative, the above rule is reversal in sign.

A. When a = 1. Solving quadratic equation: x^2 + bx + c = 0.

When a =1, solving this quadratic equation by the new method is simple, fast, and doesn't requires factoring

Example 1. Solve: x^2 - 21x - 72 = 0.

Solution. Rule of signs shows the two roots have opposite signs.

Write factors-sets of c = -72: (-1, 72) (-2, 36) (-3, 24) Stop! This sum is 21 = -b.

The 2 real roots are: -3 and 24. No factoring!

Note. There are factors-sets in opposite sign (1, -72)(2, -36)...but they can be ignored since they give opposite diagonal sums. By convention, always put the negative sign in front of the first number.

Example 2. Solve: x^2 - 39x + 108 = 0.

Solution. Both roots are positive. Write factors-sets of c = 108.

They are: (1, 108) (2, 54) (3, 36)...Stop! This sum is 3 + 36 = 39 = -b.

The 2 real roots are: 3 and 36.

Example 3. Solve: x^2 + 27x + 50 = 0.

Solution. Both roots are negative. Factors-sets of c = 50: (-1, -50)(-2, -25)...Stop!. This sum is -27 = -b. The real roots are: -2 and -25.

B. When a and c are prime/small numbers.

The new method 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.

Example 4. Solve: 7x^2 + 90x - 13 = 0.

Solution. Roots have opposite signs. Write the c/a setup. The numerator contains unique factors pair of c = -13: (-1, 13). The denominator contains unique factor pair of a = 7 that is always kept positive: (1, 7). There is unique probable root pair: (-1/7 & 13/1). The other pair can be ignored since 1 is not a real root. The diagonal sum of the unique pair is -1 + 91 = 90 = b. According to the Rule for the diagonal sum, when the diagonal sum equals b, the real roots are: 1/7 and -13

Example 5. Solve: 7x^2 - 57x + 8 = 0.

Solution. Both roots are positive. Constant c = 8 has 2 factors pairs (1, 8), (2, 4). The c/a setup: (1, 8),(2, 4)/(1, 7) leads to 3 probable root pairs: (1/7 & 8/1) (2/1 & 4/7)(2/7 & 4/1). The diagonal sum of first set is:1 = 56 = 57 = -b. The real roots are 1/7 and 8.

Example 6. Solve: 6x^2 - 19x - 11 = 0.

Solution. Roots have opposite signs. Constant a = 6 has 2 factors-sets:(1, 6) (2, 3). The c/a setup: (1, 11)/(1, 6)(2, 3) give 3 probable roots pairs: (-1/6 & 11/1) (-1/2 & 11/3) (-1/3 & 11/2)

The second set has as diagonal sum: (22 - 3 = 19 = -b). The 2 real roots are: -1/2 and 11/3.

Note. There are opposite sign roots-pairs (1/6 & -11/1) (1/2 & -11/3)... but they can be ignored since they all give opposite diagonal sums.

C. When a and c are large numbers and contains themselves many factors.

These cases are considered complicated because the (c/a) setup contains many factor pairs in both numerator and denominator. In this case, the Diagonal Sum Method can transform a complicated multiple step solving process into a simplified one by doing some elimination operations. To know how to solve these complicated cases, please read the article "Solving complicated cases of quadratic equations" on this Wiki Answers website.

NOTE. The Diagonal Sum proceeds solving by basing on the c/a setup. That is why, this method may be called: The (c/a) Method.