This new improved formula to solve a quadratic equation in its general form: ax2 + bx + c = 0, was developed and copyrighted by Nghi H. Nguyen, a Vietnamese American math tutor. This new formula is called by the author: “The quadratic formula in graphic form” and it gives the roots of a quadratic equation by this relation:

x1 = - b/2a + d/2a   and   x2 = - b/2a – d/2a        (1)

The quantity (-b/2a) represents the x-intercept of the symmetry-axis of the parabola which is the graph of the related quadratic equation.
The quantities (+d/2a) and (–d/2a) represent the two distances from the axis of symmetry to the x-intercepts of the parabola.
The value d can be zero or a real, radical, or imaginary number that will translate to double root, two real roots, or imaginary roots.
The quantity d can be easily found from the relation about the product of the two roots:    x1. x2 = c/a.
It then gives:     d
2 = b2 – 4ac.            (2)
To solve a quadratic equation, first compute the value of d by using the relation (2) then find the roots by using the formula (1).

Compared to the classic quadratic formula, this new improved one is simpler and easier to remember since students can relate it to the graphic interpretation of the solution. In addition, the quantities (d/2a) and (-d/2a) make more sense, about distances, than the quantity containing the square root of the Discriminant D = b
2 – 4ac.

The use of the quadratic formula is obligatory whenever the related quadratic equation is un-factorable into two binomials in x. If the equation is factorable, there is a new method, called “The Diagonal Sum Method”, which can solve it faster and more convenient than the existing trial-and-error factoring method. This new method was developed and copyrighted by Nghi H Nguyen, co-authored with two San Jose, California American math teachers, Karen Kusanovich and Wendy Lawson. The new math method was explained and presented in the new math text book titled:

 “Solving math quadratic equations and inequalities” (

    In existing algebra math books, there are two common methods to solve a quadratic equation in its general form a
x2 + bx + c = 0. The first method uses the quadratic formula. The second method solves the equation by using the trial-and-error factoring method. To factor a quadratic equation, students have to mentally guess the values of four constants in two binomials in x that make the equation true. Practically, most of the students feel uncomfortable in finding the rights constants because there are many permutations involved.

The new “Diagonal Sum Method” uses an innovative concept, not developed in other math books, that is finding two fractions knowing their sum and their product. The book sets up easy instructional steps for a student, with average mental math ability, to be able to solve a quadratic equation quickly. This method offers the advantage of giving the real roots of the quadratic equation directly, without having to factor it. It is very fast and convenient when the constants a, b, c are big/odd numbers. This new method can replace the trial-and-error factoring method in Algebra classes since it is faster, more convenient and it can be applicable whenever the given quadratic equation is factorable.

    Besides the new math method in solving quadratic equations, the authors also introduce two new math methods in solving inequalities. The new methods are presented   in two math textbooks recently published and titled:

“Linear equations, inequalities, and functions” (
“Trigonometry: Solving trigonometric equations and inequalities” (

These three new math books mentioned above are being sold online at the publisher’s website and on To find them, log into the publisher websites,, Yahoo or Google, then type in the author’s name or the book’s title.

The first new copyrighted method for solving a system of linear/quadratic inequalities in one variable introduces an innovative number line technique. The existing method, taught in worldwide high school algebra classes, tells students to picture the combined solution set of both inequalities on the same number line by drawing confusing cross marks in two different directions. The new method, using a double number line and picturing separately each solution set on its given number line, will help students easily visualize the combined solution set of the system through superimposing technique. A triple number line may be used to find the combined solution for a system of three inequalities.

The second new method introduces graphing of functions as an educated way for solving any inequalities and systems of inequalities. Graphing methods have always been more convenient than algebraic methods in math problem solving since students can visualize the solution. In this direction, the books develop a graphic method to solve any inequalities and systems of inequalities in one variable by considering the relative position of the function’s graph with respect to the x-axis. The consequence of this innovative method is amazingly useful. High school students, after learning this approach and with the help of graphing calculators, will be able to solve various types of inequalities and complex systems of inequalities in one variable much more easily than using algebraic manipulation. Besides linear and quadratic forms, various inequalities in one variable may include cubic, bi-quadratic, rational, radical, exponential and trigonometric inequalities. This new method which consists of reading graphs, analysis and making conclusions would fulfill the true purpose of high school math teaching./.

(by Nghi H Nguyen)

(Please click to read pdf file)