• How to solve quadratic equations by the new Diagonal Sum Method

SOLVING QUADRATIC EQUATIONS BY THE NEW DIAGONAL SUM METHOD.

  • Beyond the 4 known existing solving methods (quadratic formula, factoring, completing the square, and graphing) there is a new method called The Diagonal Sum Method that can quickly and directly give the 2 real roots, if the given equation can be factored. If this method fails, then the quadratic formula must be applied. It is the fastest way to know if a given quadratic equation can or can't be factored.
  • INNOVATIVE CONCEPT OF THE 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 SIGN FOR REAL ROOTS.
  • By this rule, students know in advance the signs of the 2 real roots (+ or -) before proceeding.
  • - If a and c have opposite signs, then the 2 real roots have opposite signs. Example1. The equation: 5x^2 -14x - 3 = 0 has 2 real roots that have opposite signs.
  • - If a and c have same sign, the 2 real roots have same sign and it can be further possible to know if both are positive and negative.
   *a. If a and b have opposite signs, both real roots are positive. Example 2: The equation x^2 - 39x + 108 has 2 real roots both positive.
   *b. If a and b have same sign, both real roots are negative. Example 3: The equation x^2 + 27x + 50 - 0 has 2 real roots both negative.
  • THE DIAGONAL SUM OF A PAIR OF 2 REAL ROOTS.
  • Given a pair of 2 real roots in the form of 2 fractions: (c1/a1 , c2/a2). Their product is (c/a) with (c1*c2 = c; a1*a2 = a). Their sum is (-b/a).
  • The sum: c1/a1 + c2/a2 = (c1a2 + c2a1)/a1a2 = (c1a2 + c2a1)/a = -b/a. The sum c1a2 + c2a1 is called the Diagonal Sum of a root-pair
  • RULE FOR THE DIAGONAL SUM.
  • The diagonal sum of a true root-pair must equal to (-b). If it equals to (b) then it is the negative of the solution. If a is negative, the above rule is reversal in sign.
  • SOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD.
  • Students select probable root-pairs from the (c/a) setup and in the same time applying the Rule of Signs. Then they use mental math to quickly calculate the diagonal sums of these probable root pairs and find the one that fits. Through practice and experiences, they will find the answers, quickly and comfortably.
  • A. When a =1, Solving quadratic equation type x^2 + bx + c = 0"'.
  • In this case, the diagonal sum reduces to the sum of the 2 real roots. Solving is simple and doesn't need factoring.
  • Example 4. Solve x^2 - 21x - 72 = 0. Solution. Rule of signs indicates roots have opposite signs. Write factor pairs of c = -72.
  • They are (-1, 72), (-2, 36), (-3, 24)...Stop. The sum of the 2 real roots in this set is 21 = -b. The 2 real roots are -3 and 24.
  • Example 5. Solve x^2 - 39x + 108 = 0. Solution. Both roots are positive. Write the factor-sets of c = 108.
  • They are (1, 108), (2, 54) (3, 36)..Stop. The sum in last set is 39 = -b. The 2 real roots are 3 and 36.
  • Example 6. Solve -x^2 - 26x + 56 = 0. Solution. Roots have different signs. a is negative. Write factor-sets of c = 56. They are: (-1, 56), (-2, 28)...Stop. This sum is 26 = -b. According to the diagonal sum rule, since a is negative, the true root-set should be the negative of this set. The 2 real roots are 2 and -28.
  • B. When a and c are prime/small numbers"'.
   *a. When a and c are both prime numbers. In this case, the number of probable root-sets is usually limited to 1.
  • Example 7. Solve 7x^2 + 90x - 13 = 0. Solution. Roots have opposite signs. Select probable root-pairs. The numerator contains unique factor-pair of c (-1, 13). The denominator contains unique factor-pair of a (1, 7). There is unique root-pair: (-1/7 , 13/1). Its diagonal sum is: -1 + 91 = 90 = b. The solution is the negative of the set. The 2 real roots are 1/17 and -13.
  • Note: The other root-pair (-1/1 , 13/7) can be ignored since 1 is not a real root.
  • Example 8. Solve 17x^2 + 324x + 19 = 0. Solution. Both roots are negative. There is unique root-set: (-1/17 , -19/1). Its diagonal sum is: -323 - 1 = -324 = -b. The 2 real roots are -1/17 and -19.
   *b. When a and c are small numbers, and contain themselves one or 2 factors. Students can write down all probable root-pairs and use mental math to calculate all diagonal sums in order to find the one that fits.
  • Example 9. Solve: 7x^2 - 57x + 8 = 0. Solution. Both roots are positive. c = 8 has 2 factor-pairs (1, 8) and (2, 4). Write all probable root-pairs: (1/7 , 8/1)  ; (2/1 , 4/7)  ; (2/7 , 4/1). Diagonal sum of first pair: 57 = -b. The 2 real roots are 1/7 and 8.
  • Example 10. Solve 6x^2 - 19x - 11 = 0. Solution. Roots have opposite signs. Probable root-pairs: (-1/6 , 11/1); (-1/2 , 11/3) ; (-1/3 , 11/2). Diagonal sum of second pair: 22 - 3 = 19 = -b. The answers are -1/2 and 11/3.

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

  • See the continuation on next article: "How to solve complicated quadratic equations using the Diagonal Sum Method" on this website.
  • (This article was written by Nghi H Nguyen, co-author of the new Diagonal Sum Method for solving quadratic equations)
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