Completing the Square

"Completing the Square" is where you ... 

... take a Quadratic Equation like this:		and turn it into this:
ax2 + bx + c = 0		a(x+d)2 + e = 0

For those of you in a hurry, I can tell you that:
and:

But if you have time, let me show you how to "Complete the Square" yourself.

Completing the Square

Say we have a simple expression like x² + bx. Having x twice in the same expression can make life hard. What can we do?

Well, with a little inspiration from Geometry we can convert it, like this:

As you can see x² + bx can be rearranged nearly into a square ...

... and we can complete the square with (b/2)²

In Algebra it looks like this:

x2 + bx		+ (b/2)2	=	(x+b/2)2
		"Complete the Square"

So, by adding (b/2)² we can complete the square.

And (x+b/2)² has x only once, which is easier to use.

Keeping the Balance

Now ... you can't just add (b/2)² without also subtracting it too! Otherwise the whole value would change.

So I will show you how to do it properly with an example:

Start with:
	("b" is 6 in this case)
Complete the Square:	Also subtract the new term
Simplify it and we are done.

The result:

x² + 6x + 7   =   (x+3)² - 2

And now x only appears once, and your job is done!

Now, let us look at a useful application: solving Quadratic Equations ...

Solving General Quadratic Equations by Completing the Square

We can complete the square to solve a Quadratic Equation (find where it is equal to zero).

But a general Quadratic Equation can have a coefficient of a in front of x²:

ax² + bx + c = 0

But that is easy to deal with ... just divide the whole equation by "a" first, then carry on:

x² + (b/a)x + c/a = 0

Steps

Now we can solve a Quadratic Equation in 5 steps:

• Step 1 Divide all terms by a (the coefficient of x²).
• Step 2 Move the number term (c/a) to the right side of the equation.
• Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

We now have something that looks like (x + p)² = q, which can be solved rather easily:

• Step 4 Take the square root on both sides of the equation.

• Step 5 Subtract the number that remains on the left side of the equation to find x.

Examples

I have two examples to show you how:

Example 1: Solve x² + 4x + 1 = 0

Step 1 can be skipped in this example since the coefficient of x² is 1

Step 2 Move the number term to the right side of the equation:

x² + 4x = -1

Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation.

(b/2)² = (4/2)² = 2² = 4

x² + 4x + 4 = -1 + 4

(x + 2)² = 3

Step 4 Take the square root on both sides of the equation:

x + 2 = ±√3 = ±1.73 (to 2 decimals)

Step 5 Subtract 2 from both sides:

x = ±1.73 – 2 = -3.73 or -0.27

And here is an interesting and useful thing. At the end of step 3 we had the equation: (x + 2)2 = 3 It gives you the vertex (turning point) of x2 + 4x + 1: (-2, -3)

Example 2: Solve 5x² – 4x – 2 = 0

Step 1 Divide all terms by 5

x² – 0.8x – 0.4 = 0

Step 2 Move the number term to the right side of the equation:

x² – 0.8x = 0.4

Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation:

(b/2)² = (0.8/2)² = 0.4² = 0.16

x² – 0.8x + 0.16 = 0.4 + 0.16

(x – 0.4)² = 0.56

Step 4 Take the square root on both sides of the equation:

x – 0.4 = ±√0.56 = ±0.748 (to 3 decimals)

Step 5 Subtract (-0.4) from both sides (in other words, add 0.4):

x = ±0.748 + 0.4 = -0.348 or 1.148

Why "Complete the Square"?

Why would you want to complete the square when you can just use the Quadratic Formula to solve a Quadratic Equation?

Well, one reason is given above, where the new form not only shows you the vertex, but makes it easier to solve.

There are also times when the form ax² + bx + c may be part of a larger problem and rearranging it asa(x+d)² + e makes the solution easier, because x only appears once.

For example "x" may itself be a function (like cos(z)) and rearranging it may open up a path to a better solution.

Also Completing the Square is the first step in the Derivation of the Quadratic Formula

Just think of it as another tool in your mathematics toolbox.

Completing the Square

By Zaib Abbasi 03:08

Derivation of Quadratic Formula

A Quadratic Equation looks like this:
                                      
A Qaudratic Formula lookd like as
                                        
The Quadratic Formula looks like a magic,but we can can follow these steps to see how it comes

1. Complete the Square

It is hard to handle an equation where "x" appears twice, but there is a way to rearrange it so that "x" only appears once. It is called "Completing the Square" (please read that first!).

So, let's go:

Start with
Divide the equation by a
Put c/a on other side
Add (b/2a)2 to both sides
The left hand side is now in the x2 + 2dx + d2 format, where "d" is "b/2a" So we can re-write it this way:
"Complete the Square"

2. Now Solve For "x"

Now we just need to rearrange the equation to leave "x" on the left

Start with
Square root
Move b/2a to right
That is actually solved! But let's simplify it a bit:
Multiply right by 2a/2a
Simplify:
Which is the Quadratic formula we all know and love:

Derivation of Quadratic Formula

By Zaib Abbasi 23:02

Quadratic Equations

The name of  the this equations i.e Quadratic comes from "Quads" which means square because variable gets squared (like x²).
It is also known as Equation of Degree 2" (because of the "2" on the x)

Here is the Standard Form of a Quadratic Equation looks like this:

                                                

a, b and c are known values. a can't be 0.

"x" is the variable or unknown (you don't know it yet)

An example of a Quadratic Equation:

                                        
Here we are giving some more Examples

          In this one a=2, b=5 and c=3

                   This one is a little more tricky:
                                        

• Where is a? In fact a=1, as we don't usually write "1x²"
• b = -3
• And where is c? Well, c=0, so is not shown.

     Oops! This one is not a quadratic equation, because it is missing x² (in other words a=0, and that means it can't be quadratic)

Quadratic Equations

By Zaib Abbasi 23:01

The Equilibrium Constant

The equlilbrium constant, K, is an expression that gives us a ratio of the products and reactants of a reaction at equilibrium with respect to a specific unit. An equilibrium constant expression is based on a reaction that is at equilibrium. There are two different types of equilibria reactions; one is heterogeneous and one is homogeneous

The Equilibrium Constant

By Zaib Abbasi 23:02

Definition of Dynamic Equilibrium

Dynamic equilibrium is the state of a reversible reaction where the forward reaction rate is equal to the reverse reaction rate, resulting in no observable net change in the system. Reactions are continuing to proceed in the forward and reverse direction dynamically; however, there is no net change in the amount of product or starting material. Dynamic equilibrium is also called steady state. In the early stage of a reversible reaction, the forward reaction proceeds more quickly than the reverse until the equilibrium state is reached when the forward and reverse rates are equal. An example of an important dynamic equilibrium is the equilibrium between ice and water at 0 ºC, where ice is melting at the same rate as the water is freezing.

Definition of Dynamic Equilibrium

By Zaib Abbasi 23:09

Reversible and Irreversible Reactions

Introduction

 The world believed that all chemical reactions were irreversible until 1803 when French chemist Claude Louis Berthollet introduced the concept of reversible reactions.

Irreversible Reactions

 As children, we learned that chemical reactions occurred when reactants reacted with each other to form products. These unidirectional reactions are known as irreversible reactions. In other words, irreversible reactions are reactions where the reactants convert to products and where the products cannot convert back to the reactants. These reactions are essentially like cooking eggs. Cooking the eggs is an  example of irreversible reaction,It does not matter how we cook the eggs,when we have cooked eggs we can not bring it back to uncooked postion

As stated above, a egg cannot be "uncooked" and so the reaction is irreversible.

A real-life example of an irreversible reaction is combustion. Combustion usually involves the burning of an organic compound, like a hydrocarbon, and oxygen to produce carbon dioxide and water. Since water is stable in its polyatomic state, like below, it will not react with the other product, CO₂, to form the reactants. Combustion can take the following form:

C_xH_y + O₂ → CO₂ + H₂O

Reversible Reactions

 In reversible reactions, the reactants and products are never used up. In fact, they are both constantly reacting and being produced. A reversible reaction can take the following summarized form:

 his reversible reaction can be broken into two reactions.

Reaction 1: 

A+B−→k1C+D

Reaction 2: 

C+D−→−k−1A+B

These two reactions are occurring simultaneously, which means that the reactants are reacting to yield the products, as the products are reacting to produce the reactants. Collisions of the reacting molecules cause chemical reactions in a closed system. After products are formed, the bonds between these products are broken because the molecules collide with each other, producing sufficient energy needed to break the bonds of the product and reactant molecules. 

Below is an example of the summarized form of a reversible reaction and a breakdown of the reversible reaction N₂O₄ ↔ 2NO₂

Reaction 1 and Reaction 2 happen at the same time because they are in a closed system.

Blue: Nitrogen    Red: Oxygen 

                                             Reaction 1                                      Reaction 2

 

Imagine a ballroom. Let reactant A be 10 girls and reactant B be 10 boys. As each girl and boy goes to the dance floor, they pair up to become a product. Once five girls and five boys are on the dance floor, one of the five pairs breaks up and moves to the sidelines, becoming reactants again. As this pair leaves the dance floor, another boy and girl on the sidelines pair up to form a product once more. This process continues over and over again, representing a reversible reaction. 

Unlike irreversible reactions, reversible reactions lead to equilibrium because reversible reactions have the reaction proceeding in both directions while irreversible reactions only have the reaction proceeding in one direction.

 If the reactants are being made at the same rate as the products are being made, a dynamic equilibrium exists. For example, if a water tank is being filled with water at the same rate as water is leaving the tank (through a hypothetical hole), the amount of water remaining in the tank remains consistent.

Reversible and Irreversible Reactions

By Zaib Abbasi 22:37

Law of Mass Action

Law stating that the rate of any chemical reaction is proportional to the product of the masses of the

reacting substances, with each mass raised to a power equal to the coefficient that occurs in the chemical equation. This law was formulated over the period 1864–79 by the Norwegian scientists Cato M. Guldberg and Peter Waage but is now of only historical interest. This law was useful for obtaining the correct equilibrium equation for a reaction, but the rate expressions it provides are now known to apply only to elementary reactions. By the term active mass we mean the molar concentration,or number of moles per dm3 in a dilute solution.


a A + b B --> c C + d Dwhere a, b, c, d are the coefficients for a balanced chemical equation.

The mass action law states that if the system is at equilibrium at a given temperature, then the following ratio is a constant.

[C]c [D]d ------------- = Keq [A]a [B]b

The square brackets "[ ]" around the chemical species represent their concentrations. This is the ideal law of chemical equilibrium or law of mass action.The units for K depend upon the units used for concentrations. If M is used for all concentrations, K has units

M^c+d-(a+b)

Accordin to law of Mass Action

the rate of forward reaction is proportional to [A] [B],while the rate of reserve reaction is equal to kr [C] [D] where kf and kr are the rate of constant for the forward and reverse reactions,respectively

Law of Mass Action

By Zaib Abbasi 10:13