• # How to solve complicated cases of quadratic equations by the diagonal sum method

SOLVING COMPLICATED QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD.

• GENERALITIES.
• Given a quadratic equation in standard form ax^2 + bx + c = 0. Solving it may be simple or complicated depending on the values of the coefficients and constant a, b, and c. This article presents various cases of solving quadratic equations by the new Diagonal Sum Method. You may see details of this new method under the article titled:"Solving quadratic equations by the Diagonal Sum Method" posted on this The Student Room website.
• A. When a = 1 - Solving quadratic equation type x^2 + bx +c = 0.
• When a = 1, solving a quadratic equation results in finding 2 numbers knowing their sum (-b) and their product (ac). The new method's solving approach is simple and fast. It doesn't require neither factoring by grouping nor solving the 2 binomials.
• Example 1. Solve: x^2 - 11x - 102 = 0. Solution. Roots have different signs. Compose factor pairs of c = -102, and in the same time, applying the Rule of Sign. By convention, always put the negative (-) sign in front of the first number of the pair. This won't affect the final result at all.
• Proceeding: (-1, 102)(-2, 51)(-3, 34)(-6, 17). OK! This last sum is (-6 + 17) = 11 = -b. The 2 real roots are -6 and 17.
• Example 2. Solve: x^2 - 28x + 96 = 0. Solution. Both real roots are positive. Compose factor pairs of c = 96 with all positive numbers. Proceeding:(1, 96)(2, 48)(3, 32)(4, 24). OK. The last sum is (4 + 24) = 28 = -b. The 2 real roots are: 4 and 24. No factoring and solving binomials.
• B. When a is not 1 - Solving equation type ax^2 + bx + c = 0.
• a. When a and c are prime number. In this case, there usually is an unique probable root pair. Solving is simple and fast. No factoring and solving binomials.
• Example 3. Solve: 7x^2 - 76x - 11 = 0. Roots have different signs. There is unique root pair: (-1/7 , 11/1). Its diagonal sum is -1 + 77 = 76 = -b. The 2 real roots are -1/7 and 11.
• b. When a and c are small numbers and may contain themselves one or 2 factors. The Diagonal Sum Method first composes the probable root pairs basing on the c/a setup. Then, it calculates their diagonal sum and find the one that fits.
• Example 4. Solve: 8x^2 - 22x - 13 = 0. Roots have different signs. There are 3 probable root pairs: (-1/8 , 13/1),(-1/2 , 13/4),(-1/4 , 13/2). The second diagonal sum is: -4 + 26 = 22 = -b. The 2 real roots are: -1/2 and 13/4.
• c. When a and c are large numbers and may contain themselves many factors.
• This case is considered complicated because there are many permutations involved. The new method first writes down the c/a setup. Then, it performs a few operations of elimination to simplify the setup. Finally, it composes probable root pairs from the remainder setup c/a.
• Example 5. Solve: 12x^2 + 5x - 72 = 0. Solution. Roots have different signs. Write down the (c/a) setup. Proceeding:
• Numerator - all factors of c = -72: (1, -72)(-2, 36)(-3, 34)(-4, 18)(-6, 12)(-8, 9).
• Denominator: all factors pairs of a = 12: (1, 12)(2, 6)(3, 4)
• Now, look to find out the pairs that don't fit. Since b is an odd number (b = 5), eliminate the pairs: (-2, 36)(-4, 18)(-6, 12) and ((2, 6)

because they give even diagonal sums. Next, eliminate the pairs: (-1, 72)(-3, 24)/(1, 12) because they give large diagonal sums while b = 5.

• The remainder c/a: (-8, 9)/(3, 4) gives 2 probable root pairs: (-8/3 , 9/4) and (-8/4 , 9/3). The first diagonal sum is 27 - 32 = -5 = -b. The 2 real roots are -8/3 and 9/4.
• Example 6. Solve" 24x^2 + 59x + 36 = 0. Solution. Both roots are negative. Write down the (c/a) setup. Proceeding:
• Numerator: (-1, -36((-2, -18)(-3, -12)(-4, -9)(-6, -6).
• Denominator: (1, 24)(2, 12)(3, 8)(4, 6).
• First, eliminate the pairs: (-2, -18)(-6, -6)/(2, 12)(4, 6) because they give even number diagonal sums, while b is odd.
• Then, eliminate the pairs:(-1, -36)(-3, -12)/(1, 24) because they give large diagonal sums, while b = 59. The remainder c/a is: (-4, -9)/(3, 8).

This gives 2 probable root pairs: (-4/3 , -9/8) and (-4/8 , -9/3). The first diagonal sum is: -32 - 27 = -59 = -b. The 2 real roots are: -4/3 and -9/8.

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