What is the formula and methods for solving quadrilateral equations? Please Help me...

There are several
methods:

1.  factoring

2. 
using the Quadratic Formula

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}.

The Quadratic
Formula -

The Quadratic Formula
is...

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 class="AM">`+-` sqrt(3^2 - 4*6*-30)] /
2*6

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

x = [-3 class="AM">`+-` 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

src="/jax/includes/tinymce/jscripts/tiny_mce/plugins/asciisvg/js/d.svg"
sscr="-4,4,-40,5,1,5,1,1,5,300,200,func,6x^2+3x-30,null,0,0,,,black,1,none"/>

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.

A note about solutions to quadratic
equations -

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:

type="image/svg+xml"
src="/jax/includes/tinymce/jscripts/tiny_mce/plugins/asciisvg/js/d.svg"
sscr="-3,5,-1,7,1,1,1,1,1,300,200,func,3x^2-5x+6,null,0,0,,,black,1,none"/>    
No solution.

type="image/svg+xml"
src="/jax/includes/tinymce/jscripts/tiny_mce/plugins/asciisvg/js/d.svg"
sscr="-10,1,-3,3,1,1,1,1,1,300,200,func,x^2+12x+36,null,0,0,,,black,1,none"/>    
One solution.

type="image/svg+xml"
src="/jax/includes/tinymce/jscripts/tiny_mce/plugins/asciisvg/js/d.svg"
sscr="-10,1,-4,3,1,1,1,1,1,300,200,func,x^2+13x+40,null,0,0,,,black,1,none"/>    
Two solutions.