What is the best method to solve quadratic equations?

A quadratic equation in one variable has as standard form: ax^2 + bx + c = 0.

There are five existing methods to solve quadratic equations: formula, factoring, completing square, graphing and the recent Diagonal Sum Method. (Amazon e-books 2010). So, students are sometimes confused about choosing the best method to proceed.

The first obvious choice is using the quadratic formula since it only requires a simple calculation to get the answer, especially when we can use calculators. But, the goal of learning math is to improve logical thinking and deductive reasoning. That is why, the math teaching process wants students to learn a few other methods, such as the factoring one, to master the solving process.

From my experiences, although 99% (?) of the quadratic equations in real life can not be factored, most of the ones given to students as exercises/problems in books or tests/exams are FACTORABLE!!!

So, before proceed solving you'd better find out if the given equation can be factored? How? In general, it is hard to know at first view if a quadratic equation is factorable. You may calculate the Discriminant D = b^2 - 4ac to see if it is a perfect square. Or, I advise you to use the Diagonal Sum Method to solve it in the first step. It is the fastest and best way to know if the equation can be factored. It usually takes fewer than 3 trials. If it fails to get answer, then the quadratic formula must be used. Here is how this new method works:

Concept of the Diagonal Sum Method.

Direct finding two real roots, in the form of two fractions, knowing their sum (-b/a) and their product (c/a).

The Rule of Sign for Real Roots:

If a and c have opposite signs, the 2 real roots have opposite signs

If a and c have same sign, both roots have same sign:

a. If a and b have opposite signs, both roots are positive.

b. If a and b have same sign, both roots are negative.

Development of the Diagonal Sum Method.

Directly select the probable root-sets, in the form of 2 fractions, that are factor-sets of c and a. The numerators are factor-sets of c. The denominators are factor-sets of a.

Product of roots-set: ( c1/a1) . (c2/a2) = c/a.

Sum of roots-set: (c1/a1) + (c2/a2) = (c1a2 + c2a1)/a1a2 = - b/a.

The sum (c1a2 + c2a1) is called the diagonal sum of the roots-set. Always use mental math to compute the diagonal sum.

Rule for the Diagonal Sum.

The diagonal sum of a TRUE roots-set must be equal to (-b). If it is equal to (b), then the answer is the opposite. If a is negative, the above rule is reversal in sign.

Advantages of the Diagonal Sum Method:

1. The Rule of Sign reduces the number of permutations in HALF as compared with the factoring method. It shows in advance the signs of the 2 roots before proceeding (opposite signs, both are positive, or both are negative).

2. It directly gives the 2 real roots WITHOUT factoring.

3. It sets up simple proceeding steps so that average students can easily perform.

4. It runs perfectly in case a is negative. Just reverse the rule of signs for real roots.

5. Smart students can quickly perform by using mental math. From my experiences, smart students can usually solve quadratic equations in less than 20 seconds!!!

6. In special case when a =1, the solving process becomes very simple. No needs of factoring!!!

7. In complicated cases when a and c are big numbers and contain themselves many factors this new method proceeds with an all-options-line so that no probable roots-sets are omitted. The elimination process then reduces a multi-solving steps-problem to a simplified one.