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Solving quadratic equations ax^2 + bx + c = 0. Beyond the 4 existing solving methods, there is a new method called the Diagonal Sum Method that can directly give the 2 real roots.

Concept of the method: Direct finding 2 real roots, in the form of 2 fractions, knowing their sum (-b/a) and product (c/a). The solving process uses 2 rules: the Rule of signs for real roots, and the Rule of the Diagonal Sum

Recall the Rule of Signs. If a and c have opposite signs, the 2 real roots have opposite signs. If a and c have same sign, the 2 real roots have same sign.

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

Examples: The equation x^2 - 21x - 72 = 0 has 2 real roots with opposite signs. The equation x^2 - 39x + 108 = 0 has 2 real roots both positive. The equation x^2 + 27x + 50 = 0 has 2 real roots both negative.

Rule of the Diagonal Sum. Given a pair of 2 real roots (1/2 , 3/4). Its diagonal sum is 1.4 + 2.3 = 4 + 6 = 10. The equation it justifies is 4x^2 - 10x + 8 = 0.

From there comes the Rule: The diagonal sum of a true root pair must equal to (-b). If it equals (b), then it is the negative of the solution.

SOLVING PRACTICES.

A. If a = 1, solving x^2 + bx + c = 0. Solving is very fast, no needs for factoring!

Example 1. Solve: x^2 - 39x + 108 = 0. Rule of signs indicates both roots are positive. Write factor-pairs of c = 108: (1, 108), (2, 54), (3, 36)...Stop! This sum is 3 + 36 = 39 = -b. The 2 real roots are 3 and 36.

Example 2. Solve x^2 - 21x - 72 = 0. Roots have opposite signs. Write factor-pairs of c = -72: (-1, 72), (-2, 36) (-3, 24)...Stop! This sum is 24 - 3 = 21 = -b. The 2 real roots are -3 and 24.

B. When a and c are prime/small numbers. The new method directly select the probable root-pairs based on the values of a and c. If both a and c are prime numbers, the number of root pairs is usually limited to one (if 1 or -1 is not a real root).

Example 3. Solve 7x^2 + 90x - 13 =0. Roots have opposite signs. Unique probable root pair since 1 is not a real root: (-1/7 , 13/1). Its DS is -1 + 90 = 90 = b. The answer is the negative of this pair, the 2 real roots are 1/7 and - 13.

When a and c are small numbers and may contain themselves a factor, the number of probable real roots is usually fewer than 3.

Example 4. Solve: 6x^2 - 19x - 11 = 0. Roots have opposite signs. Constant a = 6 has 2 factor-pair (1, 6), (2, 3).

There are 3 probable root pairs: (-1/6, 11/1) (-1/2, 11/3) (-1/3, 11/2). DS of the second pair: 19 = -b. The 2 real roots are -1/2 and 11/3.

C. When a and c are large numbers and may contain themselves many factors, solving becomes complicated since it involves many permutations. However, we can proceed the elimination of non-fitted probable root-pairs. The remainder option is usually fewer than 3 trials. To do not omit any root pair, we create an all options setup of (c/a). The numerator of the setup contains all factor-pairs of c. The denominator contains all factor pairs of a. Then we proceed elimination of non-fitted options. With practices and experiences, we can easily find the option that fits.

Example 5. Solve 8x^2 + 13x - 6 = 0. Roots have opposite signs. Write all option setup

Numerator: (-1, 6) (2, 3)

Denominator:(1, 8) (2, 4).

First eliminate the option that link to pair (2, 4) because it give even-number diagonal sum (while b is odd). Next, eliminate the option linked to pair (-1, 6) since it give larger diagonal than b = 13. The remainder option (-2, 3)/(1, 8) gives 2 probable root pairs (-2/1 , 3/8) and (-2/3 , 3/1). The first pair has as DS -16 + 3 = -13 = -b. The 2 real roots are -2 and 3/8.

Example 6. Solve 12x^2 - 272x + 45 = 0. Both roots are positive. Write all option setup:

Numerator: (1, 45)(3, 15)(5, 9)

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

First, eliminate options linked to pairs (1/12),(3/4)because they give odd-number diagonal sums while b is even. Next, look for a large DS (272). The fitted option should be the option (1, 45)/(2,6) that gives the 2 probable root pairs (1/2, 45/6),(1/6, 45/2). The second pair gives 2 real roots 1/6 and 45/2.

Advantages of the diagonal sum method. It is fast. It directly gives the 2 real roots. It saves the time used to solve the 2 binomials. It works perfectly when a, b,c are large numbers while computation by the quadratic formula may have difficulties if calculators are not allowed, during some tests/exams for example.

To know about this new method, please read the article titled:"How to solve quadratic equations by the Diagonal Sum Method" on the WikiHow or Enotes websites.

Sorry, please correct 2 minor mistakes in the above article.

1. Line 15. Please read:"The equation that it justifies is: 8x^2 - 10x + 3 = 0" (instead of 4x^2 - 10x + 8 = 0).

2. Line 28. Please read:"Its diagonal sum is -1 + 91 = 90 = -b" (instead of -1 + 90 = 90 = -b)

Thu144

Solving quadratic equations has been one main core subject of high school math. Studies on quadratic equation solving are numerous and various. So far, there are 7 common existing methods to solve quadratic equations in standard form ax^2 + bx + c = 0. Each of these methods has its pros and cons.

1. The graphing method only give approximate answers. The accuracy of the answers is variable and depends on how accurate you can graph the parabola. In addition, drawing a parabola graph takes too much time. Unless told to do, this method is unadvised.

2. The method of completing the squares may be simple and easy when the coefficients and constant a, b, and c are small numbers. When they are large numbers and especially when b is an odd number, guessing and completing the squares take a lot of time and usually lead to error/mistakes. In addition, the quadratic formula is itself the final product of this method, so you'd better use the formula to save time.

3. The factoring method usually works in 2 ways: guessing or systematic (The AC Method). The guessing approach can be easily done when a, b, and c are small numbers. When they are large numbers, guessing takes lot of time and easily leads to errors. The systematic AC Method has been so far the most popular one to solve quadratic equations in standard form. However, it could be considerably improved if we apply the Rule of Signs for Real Roots of a quadratic equation in its solving process. See the articles titled:"Solving quadratic equations by the new and improved Factoring AC Method" on Google and Yahoo Search. In case a = 1, equation type x^2 + bx + c = 0, this method can immediately obtain the 2 real roots without factoring by grouping and solving the 2 binomials. In general cases, this method presents the effective way to compose factor pairs of the product a*c.

4. The Bluma Method. This method presents a new idea of transformation of equation, from the one in standard form into the one in simplified form (a = 1). However, it still use the factoring method to get the 2 real roots, therefore it doesn't offer anything better than the AC Method.

NOTE 1. In general, we can divide the quadratic equations into 2 categories: those which can't be factored and those which are factorisable. We can also subdivise these categories into simple types (when a, b, and c are small numbers), or complicated types (when a, b, c are large numbers).

When the given quadratic equation can't be factored, the quadratic formula would be the obvious choice.

When the given quadratic equation can be factored, the AC method, the Diagonal Sum Method, and the new Transforming Method are the best one to select.

5. The quadratic formula. This formula can easily solve various quadratic equations, especially when calculators are allowed. However, students may sometimes feel the formula use routine, abundant and tedious. In fact, the ultimate goals of math learning are to improve logical thinking and deductive reasoning of students. That is why many quadratic equations, given in books/tests/exams are intentionally set up so that students must use other solving methods than the formula to solve them.

6. The new Diagonal Sum Method. This method directly obtains the 2 real roots, in the form of 2 fractions, knowing their sum (-b/a) and their product (c/a). It is fast, convenient and is applicable whenever the equation can be factored. It presents many advantages. When a = 1, it immediately obtains the 2 real roots, without factoring by grouping and solving the binomials. In case of complicated equations, when a, b, and c are large numbers and may contain themselves many factors, this method can transform a multiple steps solving process into a simplified one by doing a few elimination operations. To know the details, please read articles titled:"Solving quadratic equations by the new Diagonal Sum method" on Google or Yahoo Search.

7. The new Transforming Method. This may be the fastest and simplest method to solve quadratic equations that can be factored. Its strong points are; fast, simple, no guessing, systematic, no factoring by grouping and no solving binomials. To know how does it work, please read articles titled:"Solving quadratic equations by the New Transforming Method" on Google or Yahoo Search.

Solving quadratic equations by the new Transforming method.

This method uses 2 features:

1. Recall the Rule of Signs of a quadratic equation.

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

Example: The equation x^2 - 5x - 6 = 0 has 2 real roots (-1) and (6) that have different signs.

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

- When a and b have different signs, both roots are positive. Exp: x^2 - 9x + 8 = 0 has 2 roots both positive

- When a and b have same sign, both roots are negative. Exp: x^2 + 13x + 12 = 0 has 2 roots both negative (-1, -12).

2. The Diagonal Sum Method to solve quadratic equation type x^2 + bx + c = 0, when a = 1. This method immediately obtains the 2 real roots without factoring and solving the binomials. Solving results in finding 2 numbers knowing the sum (-b) and the product (c). This method proceeds by composing the factor pairs of (c) following these 3 TIPS.

TIP 1. If roots have different signs, compose factor pairs of c with all first numbers being negative.

Example 1. Solve x^2 - 31x - 102 = 0.

Roots have different signs, Compose factor pairs of c = -102. Proceeding: (-1, 102)(-2, 51)(-3, 34). This last sum is 34 - 3 = 31 = -b. Then, the 2 real roots are: -3 and 34.

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

Example 2. Solve: x^2 - 28x + 96 = 0.

Both roots are positive. Compose factor pairs of c = 96. Proceeding: (1, 96)(2, 48)(3, 32)(4, 24). This last sum is 4 + 24 = 28 = -b. Then, the 2 real roots are: 4 and 48.

TIP 3. If both roots are negative, compose factor pairs of c with all negative numbers.

Example 3. 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 : -3 - 36 = -39 = -b. Then, the 2 real roots are: -3 and -36.

NOTE. If you can't find the factor pair whose sum equals to (-b), or b, then this equation can't be factored and you should probably use the quadratic formula to solve it.

THE NEW TRANSFORMING METHOD

This new method proceeds through 3 steps.

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

STEP 2. Solve the transformed equation (2) by the Diagonal Sum Method that immediately obtains the 2 real roots; y1 and y2.

STEP 3. Back to the original equation (1), the 2 real roots are: x1 = y1/a, and x2 = y2/a.

Example 4. Solve 8x^2 - 22x - 13 = 0 (1).

Step 1, the transformed equation is: x^2 - 22x - 104 = 0 (2).

Step 2. Solve the equation (2). Roots have different signs. Proceeding: (-1, 104)(-2, 52)(-4, 26). This last sum is 26 - 4 = 22 = -b. Then, the 2 real roots are: y1 = -4, and y2 = 22.

Step 3: Back to the original equation (1), the 2 real roots are: x1 = y1/8 = -4/8 = -1/2, and x2 = y2/8 = 26/8 = 13/4.

Example 5. Solve: 24x^2 + 59x + 36 = 0 (1).

Step 1: Transformed equation x^2 + 59x + 864 = 0 (2)

Step 2. Solve equation (2). Both roots are negative. Compose factor pairs of a*c = 864 with all negative number. Start composing from the middle of the factor chain to save time. Proceeding: ...(-18, -48)(-24, -36)(-27, -32). This last sum is: -27 - 32 = -59 = -b. Then, the 2 real roots of (2) are: y1 = -27, and y2 = -32.

Step 3. Back to the original equation (1), the 2 real roots are: x1 = y1/24 = -27/24 = -9/8, and y2 = -32/24 = -4/3.

CONCLUSION. The strong points of the new Transforming Method are: fast, simple, no guessing, systematic, no factoring by grouping and no solving the binomials.

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