|

Hot goIITian

Posted on
23 Jan 2009 21:12:46 IST
Posts: 106
23 Jan 2009 21:12:46 IST
0 people liked this
3
2067
Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Algebra

quadratic function is any function equivalent to one of the form

Here are some examples of quadratic functions

quadratic equation is any equation equivalent to one of the form

Here are some examples of quadratic equations

A real number x will be called a solution or a root if it satisfies the equation, meaning  . It is easy to see that the roots are exactly the x-intercepts of the quadratic function  , that is the intersection between the graph of the quadratic function with the x-axis.

 a<0 a>0

Example 1: Find the roots of the equation

Solution. This equation is equivalent to

Since 1 has two square-roots  , the solutions for this equation are

Example 2: Find the roots of the equation

Solution. This example is somehow trickier than the previous one but we will see how to work it out in the general case. First note that we have

Therefore the equation is equivalent to

which is the same as

Since 3 has two square-roots  , we get

which give the solutions to the equation

First recall the algebraic identities

We shall use these identities to carry out the process called Completing the Square. For example, consider the quadratic function

What can be added to yield a perfect square? Using the previous identities, we see that if we put 2e=8, that is e=4, it is enough to add  to generate a perfect square. Indeed we have

It is not hard to generalize this to any quadratic function of the form  . In this case, we have 2e=b which yields e=b/2. Hence

Example: Use Complete the Square Method to solve

Solution.First note that the previous ideas were developed for quadratic functions with no coefficient in front of  . Therefore, let divide the equation by 2, to get

which equivalent to

In order to generate a perfect square we add  to both sides of the equation

Easy algebraic calculations give

which give the solutions to the equation

Many inequalities lead to finding the sign of a quadratic expression. let us discuss this problem here. Consider the quadratic function

We know that

1
if  (double root case), then we have

In this case, the function  has the sign of the coefficient a.

 a<0 a>0

2
If  (two distinct real roots case). In this case, we have

where  and  are the two roots with  . Since  is always positive when  and  , and always negative when , we get

•  has same sign as the coefficient a when  and  ;
•  has opposite sign as the coefficient a when  .
 a<0 a>0

3
If  (complex roots case), then  has a constant sign same as the coefficient a.

 a<0 a>0

Example: Solve the inequality

Solution. First let us find the root of the quadratic equation  . The quadratic formula gives

which yields x= -1 or x=2. Therefore, the expression  is negative or equal to 0 when  .

An Extra..........

Consider the series  and its associated sequence of partial sums  . We will say that  is convergent if and only if the sequence  is convergent. The total sum of the series is the limit of the sequence  , which we will denote by

So as you see the convergence of a series is related to the convergence of a sequence. Many do some serious mistakes in confusing the convergence of the sequence of partial sums  with the convergence of the sequence of numbers  .

Basic Properties.

1.
Consider the series  and its associated sequence of partial sums  . Then we have the formula

for any  .
This implies in particular that if we know sequence of partial sums  , one may generate the numbers  since we have

2.
If the series  is convergent, then we must have

In particular, if the sequence we are trying to add does not converge to 0, then the associated series is divergent.

3.
The geometric series

converges if and only if |q|<1. Moreover we have

4.
(Algebraic Properties of convergent series) Let  and  be two convergent series. Let  and  be two real numbers. Then the new series

is convergent and moreover we have

Example. Show that the series

is divergent, even though

Answer. Note that for any  , we have

Hence we have (for the associated partial sums)

Since  , then we have

which implies that the series is divergent. Indeed, we do have

since

which implies

Example. Check that the following series is convergent and find its total sum

Using the above properties, we see here that we are dealing with two geometric series which are convergent. Hence the original series is convergent and we have

which gives

Example. Check that the following series is convergent and find its total sum

Answer. First we need to clean the expression (by using algebraic manipulations)

We recognize a geometric series. Since  , then the series is convergent and we have

I'm trying to help students like me for some last revivision.

You can NUDGE me if u want help on any topic..........I'll try my best to help...Hope the article helped.

Blazing goIITian

Joined: 16 Mar 2008 20:01:16 IST
Posts: 1801
24 Jan 2009 11:06:52 IST
0 people liked this

nice one.. keep up with ur gr8 work...

Blazing goIITian

Joined: 13 Aug 2008 18:43:09 IST
Posts: 1313
24 Jan 2009 20:09:49 IST
0 people liked this

good article :)

Hot goIITian

Joined: 24 Jul 2008 15:02:55 IST
Posts: 106
25 Jan 2009 18:09:34 IST
0 people liked this

Hey guys did u all see it !!!!!!!!

 Some HTML allowed. Keep your comments above the belt or risk having them deleted. Signup for a avatar to have your pictures show up by your comment If Members see a thread that violates the Posting Rules, bring it to the attention of the Moderator Team

## For Quick Info

Name

Mobile

E-mail

City

Class

Vertical Limit

Top Contributors
All Time This Month Last Week
1. Bipin Dubey
 Altitude - 16545 m Post - 7958
2. Himanshu
 Altitude - 10925 m Post - 3836
3. Hari Shankar
 Altitude - 10085 m Post - 2217
4. edison
 Altitude - 10825 m Post - 7804
5. Sagar Saxena
 Altitude - 8635 m Post - 8064