How To: Compare methods of solving quadratic equations

Please update your teaching materials with 3 new methods:
1. The Diagonal Sum method, a new method to quickly solve a quadratic equation that can be factored
2. The Double Number Line new method to solve a system of two inequalities
3. Using graphing calculators to solve ANY inequality and any system of inequalities
See book titled "New methods for solving quadratic equations and inequalities" (Trafford Publishing 2009)
References: Campusi.com; Bookfinder4you.com

The factoring method is quick and easy when a = 1. However, when the constants a, b, c are large numbers and contain themselves many factors, then this method become complicated and consumes lot of time, due to high number of permutations.

Examples: (12x^2 - 51x - 45 = 0), (36x^2 + 37x - 48 = 0), (45x^2 -152x - 36 = 0)
In these cases, the new Diagonal Sum Method can directly give the 2 roots in the form of 2 fractions, WITHOUT having to factor the equation. This new method is applicable whenever the equation is factorable. If it fails then the quadratic formula must be used.

Solving by the quadratic formula is simple when the constants are small numbers. When they are large numbers, you may have computing problems especially when you can not use calculators during some tests/examinations. Another problem is that calculators always give answer in decimals while TRUE answers are sometimes in fractions. Exp: (x = 3/7), (x = 4/13)

From my experiences, I advise you to always proceed solving a quadratic equation in standard form ax^2 + bx + c = 0 in 2 steps. First step, find out if the equation can be factored? How? Solve it by the Diagonal Sum Method. It usually takes fewer than 3 trials. If it fails, then the equation is not factorable, and consequently, the quadratic formula must be used in the second step.

This new method's concept is finding 2 fractions knowing their sum (-b/a) and their product (c/a). The 2 real roots of a quadratic equation in standard form ax^2 + bx + c = 0 must be 2 fractions in the form (c1/a1) and (c2/a2) with c1.c2 = c and with a1.a2 = a

So, the 2 numerators of the 2 roots are 2 factors of c. The two denominators should be 2 factors of a. We can list these possible roots depending on the values of a and c.
Exp 7x^2 + 90x - 13 = 0. The 2 roots may be the fractions (-13/1) and (1/7) or (-13/7) and (1/1)

We can find the right answer by using a simple formula called The Diagonal Sum :
c1.a2 + a1.c2 = - b . In the above example the 2 roots are (-13) and (1/7) since the Diagonal Sum equals to -b = 90. This simple formula can be easily computed by mental math.

When the constants a, b, c are prime/small numbers, the Diagonal Sum Method usually requires fewer than 2 trials. It takes from 13 to 20 secondes to complete. When the constants a, c are large/even numbers and/or contain themselves many factors, this method requires fewer than 3 trials if students master the solving process.

Questions:
How to solve quadratic inequalities ? How many methods are there to solve quadratic inequalities? How many steps are there to solve a quadratic inequaliy ?

There are 3 methods to solve a quadratic inequality in standard form ax^2 + bx + c 0 (or

There are 3 methods to solve a quadratic inequality : ax^2 + bx + c 0 (or

There are 3 methods, 3 steps and 2 approaches in solving quadratic inequalities in standard form: ax^2 + bx +c 0 (or

3 METHODS FOR SOLVING QUADRATIC INEQUALITIES:
1. Substitution (check point)
2. Algebraic method
3. Graphing method

3 STEPS FOR SOLVING QUADRATIC INEQUALITIES:
1. Transform the inequality into a trinomial f (x) on the left side
2. Solve the quadratic equation f(x) = 0
3 Solve the inequality f(x), greater or less than 0.

2 APPROACHES IN SOLVING QUADRATIC INEQUALITIES:
a. Check point when using the number-line or graphing method
b. The Theorem about Sign of a Trinomial f(x) when using the algebraic method