• Solving quadratic equations by the new and improved factoring AC Method

SOLVING QUADRATIC EQUATIONS BY THE NEW AND IMPROVED FACTORING AC METHOD.

  • By Nghi H Nguyen.
  • So far, the AC Method (Google Search) has been the most popular method to solve a quadratic equation in standard form ax^2 + bx + c = 0 that can be factored.
  • THE FACTORING AC METHOD.
  • Given a quadratic equation in standard form ax^2 + bx + c = 0. This method aims to factor by grouping this equation into 2 binomials in x by replacing the term (bx) by the 2 terms (b1x) and (b2x) that satisfy these 2 conditions: 1) The product b1*b2 = ac. 2) The sum (b1 + b2) = b.
  • Example 1. Solve: 15x^2 - 53x + 16 = 0. Solution. Find 2 numbers that the product is: ac = 15*16 = 240 and the sum is: b = -53. The methods proceeds by composing all factors of ac = 240.
  • Proceeding: (1, 240)(2, 120)(3, 80)(4, 60)(5, 48).
  • (-1, -240)(-2, -120)(-3, -80)(-4, --60)(-5, -48). OK
  • Next, replace in the equation the term (-53x) by the 2 terms (-5x) and (-48x):
  • 5x^2 - 5x - 48x + 16 = 5x(3x - 1) - 16(3x -1) = (3x - 1)(5x - 16) = 0.
  • Then, solve the 2 binomials for x: (3x - 1) = 0 --> x = 1/2; and (5x - 16) = 0 --> x = 16/5.
  • NOTE. In the solving approach, in order to get the product ac = 240, students have to compose both factor pairs (1, 240) and (-1, -240), (2, 120) and (-2, -120) and so on...Is it possible to simplify the solving process if we know in advance the signs of the 2 real roots? Yes, if we apply the Rule of Signs for Real Roots of a quadratic equation into the solving process.
  • RECALL THE RULE OF SIGNS FOR REAL ROOTS.
  • a. If a and c have different signs, roots have different signs.
   *Example: The equation 8x^2 - 11x - 19 = 0 has 2 real roots with different signs: -1 and 19/8.
  • b. If a and c have same sign, both real roots have same sign.
   *1. If a and b have same sign both real roots are negative.
       *Example: The equation 8x^2 + 11x + 3 = 0 has 2 real roots both negative: -1 and -3/8.
   *2. If a and b have different signs, both real roots are positive.
       *Example: The equation 13x^2 - 21x + 8 = 0 has 2 real roots both positive: 1 and 8/13.
  • THE NEW AND IMPROVED FACTORING AC METHOD.
  • Its approach aims to factor the quadratic equation into 2 binomials in x by replacing in the equation the term (bx) by the 2 terms (b1x) and (b2x) that satisfy these 3 conditions:1)- The product b1*b2 = ac; 2)- The sum (b1 + b2) = b; 3)- Application of the Rule of Signs in the solving process.
  • A. When a = 1 - Solving quadratic equation type x^2 + bx + c = 0.
  • When a =1, solving results in finding 2 numbers knowing their sum (-b) and their product (c). In this case, if we apply the Rule of Signs, we can immediately get the 2 real roots when composing the factors of c.
  • Tips for the solving process when a = 1.
  • a. When roots have different signs (a and c different signs), always compose factors of the product (ac) with first number in the pair being negative. This convention doesn't affect the results at all.
  • Example 2. Solve: x^2 - 11x - 102 = 0. Solution. Roots have different signs. Compose factors of ac = c = -102 with all first numbers negative. Proceeding: (-1, 102)(-2, 51)(-3, 34)(-6, 17). OK! This last sum is (17 - 6) = 11 = -b. The 2 real roots are: -6 and 17. There is no need neither for factoring by grouping nor by solving the binomials.
  • Note. If we compose factors of ac = -102 differently, the outcome will be the same. Proceeding (1, -102)(2, -51)(3, -34)(6, -17). The last sum is 6 - 17 = -11 = b. Therefore, the negative of this set gives the 2 real roots. They are -6 and 17.
  • b. When roots have same sign (a and c same sign), compose factors of the product ac with all positive numbers.
  • Example 3. Solve x^2 + 31x + 108 = 0. Solution. Both real roots are negative. Proceed factors of c = 108 with all positive numbers.
  • Proceeding: (1, 108)(2, 54)(3, 36)(4, 27). OK! This last sum is 4 + 27 = 31 = b. Then, the 2 real roots are: -4 and -27. There is no need for factoring by grouping and solving the binomials.
  • Note. If we compose factors of c = 108 differently, the outcome will be the same. Proceeding: (-1, -108)(-2, -54)(-3, -36)(-4, -27). OK. This last sum is (-4 - 27) = -31 = -b. Then, the 2 real roots are -4 and -27.
  • B. When a is not 1 - Solving quadratic equation type ax^2 + bx + c = 0.

By applying the Rule of Signs into the solving process, we can proceed solving with the same 2 Tips, mentioned above.

  • Example 4. Solve: 8x^2 - 22x - 13 = 0. Solution. Roots have different signs. Compose factors of ac = -104 with all first numbers being negative.
  • Proceeding: (-1, 104)(-2, 52)(-4, 26). OK! This last sum is: (-4 + 26) = 22 = -b. Then (b1 + b2 ) = b = (4 - 26). Next, replace in the equation the term (-22x) by the 2 term (4x) and (-26x). Then factor by grouping: 8x^2 - 4x + 26x - 13 = 4x(2x+ 1) - 13(2x + 1) = 0.
  • Next, solve the 2 binomials: (2x + 1) = 0 --> x = -1/2, and (4x - 13) = 0 --> x = 13/4
  • Example 5. Solve: 12x^2 + 29x + 15 = 0. Solution. Both real roots are negative. Compose factors of ac = 180 with all positive numbers. *Proceeding: (1, 180(2, 90)(3, 60)(4, 45)(5, 36)(6, 30)(9, 20). OK. This last sum is (9 + 20) = b. Then b1 = 9 and b2 = 20. Next, replace in the equation the term (29x) by the 2 terms (9x) and (20x).
  • 12x^2 + 9x + 20x + 15 = 3x(4x + 3)+ 5(4x + 3) = (4x + 3)(3x + 5) = 0.
  • (4x + 3) = 0 --> x = -3/4; and ((3x + 5) = 0 --> x = -5/3.
  • CONCLUSION AND REMARKS.
  • The new and improved factoring AC Method helps:
  • 1. To know in advance the signs of the 2 real roots for a better selection of solving approach.
  • 2. To simplify the solving process by reducing the numbers of permutations in both cases: roots have same sign or roots have different signs.
  • Example. When roots have different signs and we compose factors of ac = -24. Instead of listing 8 pairs: (-1, 24) (1, -24)(-2, 12)(2, -12) (-3, 8)(3, -8)(-4, 6)(4 -6), we only need to list 4 pairs: (-1, 24)(-2, 12)(-3, 8)(-4, 6).
  • Example. When roots have same sign (- or +), and we compose factors of ac = 24. We only list factors with positive numbers: (1, 24)(2, 12)(3, 8)(4, 6) and ignore the factors: (-1, -24)(-2, -12)(-3, -8)(-4, -6).
  • Note. we can stop the factor listing after we find the pair that fit. This saves a lot of time.
  • 3. When a = 1, the 2 real roots can be immediately obtained from the list of the factors of c. There is no need for factoring by grouping and for solving the binomials.
  • NOTE. When composing factors of ac, if we don't find any factor sum (b1 +b2) that matches b (or -b), then the given quadratic equation can't be factored, and consequently, the quadratic formula must be used to solve it.
  • (This article was authored by Nghi H. Nguyen, the co-author of the Diagonal Sum Method for solving quadratic equations)
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