samhouston | Certified Educator

There are several methods:

1.  factoring

3.  graphing

No matter which method you choose to use, you always want to begin by rewriting the equation in standard form.  That is...

ax^2 + bx + c = 0

Factoring -

x^2 + 5x + 6 = 0

Think of two numbers whose product is c and whose sum is b.

2 * 3 = 6          2 + 3 = 5

Now rewrite the equation as a product of two binomials using these numbers.

(x + 2)(x + 3) = 0

To get a product of 0, one or both of the factors must equal 0.  Take each binomial separately, set it equal to 0, and solve.

x + 2 = 0          x = -2

x + 3 = 0          x = -3

The solution set is {-2, -3}.

x = [-b `+-` sqrt(b^2 - 4ac)] / 2a

6x^2 + 3x + -30 = 0

Identify a, b, and c.

a = 6

b = 3

c = -30

Substitute these numbers into the Quadratic Formula and solve.

x = [-3 `+-` sqrt(3^2 - 4*6*-30)] / 2*6

x = [-3 `+-` sqrt(9 - -720)] / 12

x = [-3 `+-` sqrt(729)] / 12

x = (-3 `+-` 27) / 12

Here, the formula splits in two, one using + and one using -.

x = (-3 + 27) / 12          x = 24 / 12          x = 2

x = (-3 - 27) / 12          x = -30 / 12          x = -2.5

The solution set is {2, -2.5}.

Graphing -

This method is the simpliest method if you have access to a graphing calculator.  If you don't have access to one, I have provided a link to an online graphing calculator.  This is also a method that is good for checking your answers when using either of the other methods.

Enter the equation in for y=.  Graph and adjust the window so that the x-intercept(s) are visible.

6x^2 + 3x + -30 = 0

y = 6x^2 + 3x + -30

Notice that the x-intercepts are -2.5 and 2.  These are the two solutions to the equation.  Notice that they match the solution set of the Quadratic Formula example.  This is why the graphing method is an effective tool for checking your work.

Quadratic equations can have 0, 1, or 2 solutions.  You can tell by the graph how many solutions the equation will have.  If the parabola does not intersect the x-axis, the equation has 0 solutions.  If the parabola intersects the x-axis at one place (its vextex), then the equation has 1 solution.  If the parabola intersects the x-axis at two places, then the equation has 2 solutions.

Example:

No solution.

One solution.

Two solutions.

nang1350 | Student

There are 5 existing methods with pros and cons.

1. The graphing method. It can only give approx. answers while the true answers are numbers and fractions (1/3, or -3/5...). The degree of accuracy depends on how accurate you graph the parabola. In addition, graphing a parabola takes too much time for an aswer during test/exam.

2. The method of completing the square. It works when a = 1. When a is not 1, guessing and arranging the perfect square take a lot of time, especially when a, b, c are large numbers. Since this method is considered as a consequence of the quadratic formula, and since time is limited during test/exam, it is advised that you'd better use the quadratic formula.

3. The quadratic formula. It is necessary when the equation can not be factored. It is fast when the constants a, b, c are small numbers and when you can use a calculator. However, when the constants are large numbers, you may have calculation problems, especially when you can not use calculator, during some tests/exams for example. Anothe problem with using calculators is that the answers are given in decimals while true answers are sometimes numbers or fractions. In addition, you have to learn by heart the formula that is a little hard to remember.

There is a new improved quadratic formula that is called "The quadratic formula in graphic form". It is easier to remember since it relates the 2 real roots of the equation to the 2 x-intercepts of the parabola graph. See book titled "New methods for solving quadratic equation" (Amazon e-book 2010).

4. The factoring method. It only works when the given equation can be factored. It is a trial and error method that tries to factor the given equation into 2 binomials. It then solves the 2 binomial for x. It looks simple when a =1 or when the constants a, b, c are small numbers. But when the constants are large numbers, it becomes confusing and consumes a lot of time. There is on "YOU TUBE" an interesting approach for factoring quadratic equations, called the "ac" method, that you need to know.

5. The new  method called "The Diagonal Sum Method". It is also a trial and error method, same as the factoring one. It can directly give the 2 real roots without having to factor the equation. It  works when the given equation is factorable. In fact, it can be considered as a shorcut of the factoring method.

Best Approach to solve quadratic equations.

First, see if the given equation is factorable. How? Solve it by using the new diagonal sum method. It usually requires fewer than 3 trials. If it fails, meaning no diagonal sum is equal to (b) or (-b), then the quadratic formula must be used.

nang1350 | Student

1. It directly give the answers. This method directly gives the 2 real roots in the form of 2 fractions. It is considerd as a shortcut of the factoring method. It saves the time used to solve the 2 binomials for x.

2. It is fast. From my experiences, average students can solve common quadratic equations in less than 20 secondes ! In particular, it is very fast when solving the equation x^2 + bx + c = 0 (when a = 1)

Example. Solve x^2 - 39x + 108 = 0. Rule of signs indicates both real roots are positive. Write factor-pairs of c = 108: (1, 108) (2, 54) (3, 36)...Stop! This sum is 3 + 36 = 39 = -b. The 2 real roots are 3 and 36. No factoring ! No time needded to solve the 2 binomials.

3. It reduces in half the number of permutations as compared with the factoring method by using the Rule of signs for real roots and the Rule of the diagonal sum.

4. It proceeds by steps. Average students, with patience and experiences, can easly perform.

Best approach to solve quadratic equations.

In first step, use the diagonal sum method to solve the given equation. Usually, it requires fewer than 3 trials, and less than 20 secondes.

If, it fails to get the answers, meaning there is no diagonal sum that equals (-b), then the quadratic formula must be used in solving in next step.

To know about the new method, please read the article titled:"How to solve quadratic equations by the diagonal sum method" on this Enotes website.