Lesson Lesson Title: Solving quadratic equations by the Diagonal Sum method

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There is a new method, called the Diagonal Sum Method, that can quickly give the 2 real roots if the given quadratic equation, in standard form ax^2 + bx + c = 0, can be factored. It uses a Rule of Signs for real roots and a Rule for The Diagonal Sum of a root-pair.

CONCEPT OF THE NEW DIAGONAL SUM METHOD: Direct finding 2 real roots, in the form of 2 fractions, knowing their sum (-b/a) and their product (c/a).

RECALL THE RULE OF SIGNS FOR REAL ROOTS:

a. If a and c have opposite signs, the roots have opposite signs.
      Example: The equation x^2 - 5x - 6 = 0 has 2 roots with opposite signs
   b. If a and c have same sign, both roots have same sign.
      1. 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
      2. If a and b have different signs, both real roots are positive.
         Exp: The equation x^2 - 10x + 9 = 0 has 2 real roots both positive.

THE RULE OF THE DIAGONAL SUM.

Given a pair of real roots: (c1/a1, c2/a2). 
Their product is (c1.c2)/(a1.a2) = c/a. It means: the numerators of a root-pair constitute a factor pair of c. The same, the denominators constitute a factor-pair of a.
Their sum is: (c1/a1 + c2/a2) = (c1a2 + c2a1)/a1a2 = (ds/a) = (-b/a).
The sum (c1a2 + c2a1) is called the Diagonal Sum of a root-pair. From there comes the Rule: 
"The diagonal sum of a true root-pair must equal to (-b). If it equals to (b), the answer is the opposite of the pair. If a is negative, the rule is reversal in sign".

SOLVING PROCESS

A. WHEN a = 1 - Solving the equation x^2 + bx + c = 0. 
In this case, the diagonal sum reduces to the sum of the 2 real roots, and it must equal to (-b). Solving is simple, fast, and doesn't need factoring.

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

NOTE: There are opposite pairs (1, -72)(2, -36)(3, -21)...but they can be ignored since they give opposite root-pairs. By convention, always put the negative sign (-) in front of the first number when roots have opposite signs. This doesn't affect the result at all.

Example 2. Solve: x^2 - 39x + 108 = 0. Both real roots are positive. Write factor pairs of a = 108.
They are: (1, 108)(2, 54)(3, 36)...This sum is 3 + 36 = 39 = -b. The 2 real roots are 3 and 36.

Example 3. Solve: x^2 + 26x - 120 = 0. Roots have opposite signs. Factor pairs of c = -120: (-1, 120)(-2, 60)(-3, 40)(-4, 30)...This sum is (30 - 4 = 26 = b). According to the Rule of Diagonal Sum, the answer is the opposite root pair. The 2 real roots are 4 and -30.

Example 4. Solve -x^2 - 26x + 56 = 0. Roots have opposite signs. The constant a is negative. Write factor-pairs of ac = -56. They are: (-1, 56)(-2, 28)...This sum is (28 - 2 = 26) = -b. Since a is negative, the answer is the opposite of this pair. The 2 real roots are 2 and -28.

REMARK. The Rule of Signs, that shows the signs of the real roots (+ or -) before proceeding, reduces in half the number of permutations (or test cases). This new method is simple and fast! It saves the time used for factoring and for solving the 2 binomials.

B. WHEN a and c ARE PRIME NUMBERS.

The Diagonal Sum Method directly selects the probable root-pairs from the (c/a) setup. It, in the same time, applies the Rule of Sign into these pairs.
In the case where both a and c are prime numbers, the number of probable root-pairs is limited to one, except when 1 (or -1) is one real root.

Example 5. Solve 7x^2 + 90x - 13 = 0. Roots have opposite signs. In the (c/a) setup, the numerator contains unique factor pair of (-1, 13). The denominator contains unique factor-pair of (1, 7). Permutation should be done to the denominator that is always kept positive. Probable root-pairs:
(-1/7, 13/1) (-1/1, 13/7). Since 1 (or -1) is not a real root, the second pair can be ignored and the remainder unique pair gives as diagonal sum: -1 + 91 = 90 = b. The answer is the opposite pair. The 2 real roots are 1/7 and -13.

Example 6. Solve 17x^2 + 324x + 19 = 0. Both roots are negative. There is unique root-pair: (-1/17, -19/1). Its diagonal sum is -323 -1 = -324 = -b. The 2 real roots are -1/17 and -19.

Example 7. Solve -5x^2 + 14x + 3 = 0. Roots have opposite signs. Since 1, or -1, is not a real root, there is unique root-pair: (-1/5, 3/1). Its diagonal sum is: -1 + 15 = 14 = b. Since a is negative, the 2 real roots are -1/5 and 3.

C. WHEN a and c ARE SMALL NUMBERS AND MAY CONTAIN THEMSELVES ONE (OR 2) FACTORS.

This case covers most of the quadratic equations (that can be factored) given in books/home works/tests. The new method first selects all probable root-pairs from the (c/a) setup. In this setup, the numerator contains all factor pairs of c. The denominator contains all factor pairs of a. Next, it uses mental math to calculate the diagonal sums of probable root pairs. It stops calculation when one diagonal sum equals to b (or -b).

Example 8. Solve 7x^2 - 57x + 8 = 0. Both roots are positive. In the (c/a) setup, the numerator contains 2 factor pairs: (1, 8) (2, 4). The denominator contains unique factor pair: (1, 7). There are 3 probable root-pairs: (1/7, 8/1), (2/1, 4/7), (2/7, 4/1). The first diagonal sum is 1 + 56 = 57 = -b. The 2 real roots are 1/7 and 8.

Example 9. Solve 6x^2 - 19x - 11 = 0. Roots have opposite signs. The constant a has 2 factor pairs: (1, 6),(2, 3). The constant c has unique factor-pair: (1, 11). Write all 3 probable root-pairs: 
(-1/6, 11/1),(-1/2, 11/3),(-1/3, 11/2). The second diagonal sum is : -3 + 22 = 19 = -b. The 2 real roots are -1/2 and 11/3.

Example 10. Solve 6x^2 -11x - 35 = 0. Roots have opposite signs. Write the (c/a) setup:
(-1, 35),(-5, 7)/(1, 6),(2, 3). Before proceeding, we can eliminate the pair (-1, 35) since it gives large diagonal sums, as compared to b = -11. There are 4 remainder probable root-pairs:
(-5/1, 7/6) (-5/6, 7/1) (-5/2, 7/3) (-5/3, 7/2). The forth diagonal sum is: -10 + 21 = 11 = -b. The 2 real roots are -5/3 and 7/2.

Example 11. Solve  -5x^2 - 69x + 14 = 0. Roots have opposite signs. Constant a is negative. The c/a set up contains, as numerator, 2 factor-pairs: (-1, 14)(-2, 7) and as denominator: (1, 5). Before proceeding, we can eliminate the pair (-2, 7) since it gives small diagonal sums, as compared to b = - 69. The remainder c/a setup: (-2, 7)/(1, 5) leads to unique root-pair: (-1/5, 14/1). Its diagonal sum is -1 + 70 = 69 = -b. Since a is negative, the answer is the opposite pair. The 2 real roots are 1/5 and -14.

REMARK 1. We can solve these quadratic equations by the factoring "ac method" (You Tube). It takes more time in factoring and solving the 2 binomials.

REMARK 2. If the new method fails to get the answer, meaning no diagonal sum equals to b (or -b), then the equation can not be factored, and consequently, the quadratic formula must be used for solving.

D. WHEN a and c ARE LARGE NUMBERS AND CONTAIN THEMSELVES MANY FACTORS.

These cases are considered complicated because there are many permutations involved. Examples of complicated quadratic equations: 24x^2 + 59x + 36 = 0; 12x^2 + 5x - 72 = 0; 45x^2 - 74x - 55 = 0; 12x^2 - 272x + 45 = 0. In these cases, the Diagonal Sum Method proceeds by considering the quotient (c/a). The numerator contains all factor pairs of c; the denominator contains all factor pairs of a.
Next, this method transforms a multiple steps solving process into a simplified one by doing a few elimination operations.

Example 12. Solve: 12x^2 + 5x - 72 = 0.
Solution. Roots have opposite signs. Create the (c/a) setup, with all factor pairs of c and of a:
Numerator: (-1, 72),(-2, 36),(-3, 24),(-4, 18),-(6, 12),(-8, 9).
Denominator: (1, 12),(2, 6),(3, 4).

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-number diagonal sums (while b = 5 is odd). Then, eliminate the pairs: (-1, 72),(-3, 24)/(1, 12) because they give large diagonal sums as compared to b = 5. The remainder (c/a) setup is: (-8, 9)/(3, 4) that leads to 2 probable root pairs: (-8/3, 9/4) and (-8/4, 9/3). The diagonal sum of the first pair is: 27 - 32 = -5 = -b. The 2 real roots are -8/3 and 9/4.

Example 13. Solve: 24x^2 + 59x + 36 = 0.
Solution. Both roots are negative. Create the (c/a) setup. 
Numerator: (-1, -36),(-2, -18),(-3, -12),(-4, -9),(-6, -6). 
Denominator: (1, 24),(2, 18),(3, 8),(4, 6).

First, eliminate the pairs: (-2, -18),(-6, -6) from the numerator, and the 2 pairs: (2, 12),(4, 6) from the denominator, because they give even-number diagonal sums (while b = 59 is odd). Then, eliminate the pairs: (-1, -36),(-3, -12)/(1, 24) because they give large diagonal sums, as compared to b = 59. The remainder (c/a): (-4, -9)/(3, 8) leads to probable root pairs: (-4/3, -9/8) and 
(-4/8, -9/3). The diagonal sum of the first pair is: -32 - 27 = -59 = -b. The 2 real roots are -4/3 and -9/8.

Example 14. Solve: 45x^2 + 94x + 48 = 0
Solution. Both roots are negative. The (c/a) setup:
Numerator: (-1, -48),(-2, -24),(-3, -16),(-4, -12),(-6, -8).
Denominator: (1, 45),(3, 15),(5, 9).

First, eliminate the pairs (-1, -48),(-2, -24),(-3, 16),(-4, -12)/(1,45),(3, 15) because they give large diagonal sums (while b = 94). The remainder (c/a) is (-6, -8)/(5, 9) that leads to 2 probable root pairs: (-6/5, -8/9) and (-6/9, -8/5). The diagonal sum of the first pair is: -40 - 54 = -94 = -b. The 2 real roots are -6/5 and -8/9.

NOTE. The new Diagonal Sum method proceeds by creating the (c/a) setup. For this reason this method may be called: "The c/a Method". There is another factoring method called: "The ac method" (You Tube) that deserves to be studied.

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{{{x[1,2] = (-b +- sqrt( b^2-4ac ))/(2a) }}}

{{{ 
    sqrt( int( 1/(1+x^2), dx, -infinity, infinity ) ) 
}}}

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{{{graph( 600, 400, -10, 10, -20, 20, x^2-4x-3, x-2, 3-x ) }}}

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