How do you solve complicated cases of quadratic equations?

Answer 1

The standard form of a quadratic equation in one variable is ax^2 + bx + c = 0.

There is a new solving method, called Diagonal Sum Method (Amazon e-book 2010) that can directly give the 2 real roots without factoring. It uses a Rule of Signs for Real Roots and a Rule for the Diagonal Sum. To know how to proceed with this new method, please read the article:"Solving quadratic equations by the Diagonal Sum method" on this Wiki Answers website.

Depending on the values of the constants a and c, solving quadratic equations may be simple or complicated.

A. When a= 1. Solving by the new method is simple and doesn't need factoring.

B. When a and c are prime/small numbers, the new method directly selects the probable root pairs from the (c/a) setup. Then it applies a simple formula to calculate all diagonal sums of these probable root pairs to find the real root pair.

C. Complicated cases: When a and c are large numbers and contains themselves many factors.

In these cases, students are advised to write down the (c/a) setup. The numerator of the setup contains all factors-sets of c. The denominator contains all factors-sets of a. Next, the new Diagonal Sum Method transforms a multiple step solving process into a simplified one by doing a few elimination operations.

Example 1. Solve: 45x^2 - 74x - 55 = 0.

Solution. Rule of sign indicates two roots have opposite signs. All-options-line:

Numerator. Factors-sets of c = -55: (-1, 55) (-5, 11)

Denominator. Factors-sets of a = 45: (1, 45)(3, 15)(5, 9)

Now, you may use mental math to calculate all diagonal sums and find the one that fits. However, the best way is to transform the c/a setup into its simplest form by doing a few eliminations.

First, eliminate the pairs (-1, 55)/(1, 45)(3, 15) because they give large diagonal sums, compared to b = 74. The simple remainder c/a: (-5, 11)/(5, 9) leads to the unique root pair: (-5/9 & 11/5). Its diagonal sum is -25 + 99 = 74 = -b. The two real roots are: -5/9 and 11/5.

Example 2. Solve: 12x^2 - 272x + 45 = 0.

Solution. Both roots are positive. Write down the (c/a) setup.

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

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

First, eliminate the pairs (1, 12) and (3, 4) because they give odd-number diagonal sums, while b is and even number. Next, look for a large diagonal sum (-272). The fitted (c/a) should be (1, 45)/(2, 6) that give 2 probable roots-sets: (1/2 & 45/6) and (1/6 & 45/2). The diagonal sum of the second set is 270 + 2 = 272 = -b. The 2 real roots are: 1/6 and 45/2.

Example 3. Solve: 45x^2 - 172x + 36 = 0.

Solution. Both roots are positive. Write down the (c/a) setup.

Numerator: (1, 36) (2, 18) (3, 12) (6, 6)

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

First, eliminate the pairs (1, 36) (3, 12) since they give odd-number diagonal sums (while b is even). Then, eliminate the pairs (6, 6)/(3, 15) because they give diagonal-sums that are multiples of 3. This would make the given equation be simplified by 3. The remainder probable root pairs are: (2, 18)/ (1, 45) and (2, 18)/(5, 9). The second c/a gives 2 probable real roots-sets (2/5 & 18/9) and (2/9 & 18/5). The second set has as diagonal sum: 172 = -b. The 2 real roots are 2/9 and 18/5.

Example 4. Solve 24x^2 + 59x + 36 = 0.

Solution. Both roots are negative. Write the (c/a) setup.

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) (-2, -18)/(1, 24) since they give too large diagonal sums (b = -59). The remainder c/a is (-4, -9)/(3, 8) that gives the 2 real roots: -4/3 and -9/8.

Comments. We see that the new Diagonal sum method can transform a multiple step solving process into a simplified one by doing a few elimination operations. With practices and experiences, students will feel the above operations routine and they can surely find the real root pair that fits. It may be simpler if you solve these complicated equations by the quadratic formula with a calculator. However, performing the above operations helps fulfill the goal of learning math, that is to improve logical thinking and deductive reasoning. It would be a boring hard work if you have to solve these complicated equations by the quadratic formula without a calculator, during some tests/exams for examples.