IITJEE-(Revision) Quadratic Equations......Have a look.

Hot goIITian

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23 Jan 2009 21:12:46 IST
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23 Jan 2009 21:12:46 IST
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IITJEE-(Revision) Quadratic Equations......Have a look.
Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Algebra

 

quadratic function is any function equivalent to one of the form

displaymath14

Here are some examples of quadratic functions

  • tex2html_wrap_inline16
  • tex2html_wrap_inline18
  • tex2html_wrap_inline20
  • tex2html_wrap_inline22

quadratic equation is any equation equivalent to one of the form

displaymath24

Here are some examples of quadratic equations

  • tex2html_wrap_inline26
  • tex2html_wrap_inline28
  • tex2html_wrap_inline30
  • tex2html_wrap_inline32

 

Consider the quadratic equation

displaymath22

A real number x will be called a solution or a root if it satisfies the equation, meaning tex2html_wrap_inline26 . It is easy to see that the roots are exactly the x-intercepts of the quadratic function tex2html_wrap_inline28 , 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

displaymath30

Solution. This equation is equivalent to

displaymath32

Since 1 has two square-roots tex2html_wrap_inline34 , the solutions for this equation are

displaymath36

 

Example 2: Find the roots of the equation

displaymath38

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

displaymath40

Therefore the equation is equivalent to

displaymath42

which is the same as

displaymath44

Since 3 has two square-roots tex2html_wrap_inline46 , we get

displaymath48

which give the solutions to the equation

displaymath50

 

First recall the algebraic identities

displaymath62

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

displaymath64

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 tex2html_wrap_inline70 to generate a perfect square. Indeed we have

displaymath72

It is not hard to generalize this to any quadratic function of the form tex2html_wrap_inline74 . In this case, we have 2e=b which yields e=b/2. Hence

displaymath80

 

Example: Use Complete the Square Method to solve

displaymath82

Solution.First note that the previous ideas were developed for quadratic functions with no coefficient in front of tex2html_wrap_inline84 . Therefore, let divide the equation by 2, to get

displaymath86

which equivalent to

displaymath88

In order to generate a perfect square we add tex2html_wrap_inline90 to both sides of the equation

displaymath92

Easy algebraic calculations give

displaymath94

Taking the square-roots lead to

displaymath96

which give the solutions to the equation

displaymath98

 

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

displaymath21

We know that

1
if tex2html_wrap_inline23 (double root case), then we have

displaymath25

In this case, the function tex2html_wrap_inline27 has the sign of the coefficient a.

a<0
a>0

 

2
If tex2html_wrap_inline31 (two distinct real roots case). In this case, we have

displaymath33

where tex2html_wrap_inline35 and tex2html_wrap_inline37 are the two roots with tex2html_wrap_inline39 . Since tex2html_wrap_inline41 is always positive when tex2html_wrap_inline43 and tex2html_wrap_inline45 , and always negative whentex2html_wrap_inline47 , we get

  • tex2html_wrap_inline49 has same sign as the coefficient a when tex2html_wrap_inline43 and tex2html_wrap_inline45 ;
  • tex2html_wrap_inline49 has opposite sign as the coefficient a when tex2html_wrap_inline47 .
a<0
a>0

 

3
If tex2html_wrap_inline63 (complex roots case), then tex2html_wrap_inline49 has a constant sign same as the coefficient a.

 

a<0
a>0

Example: Solve the inequality

displaymath69

Solution. First let us find the root of the quadratic equation tex2html_wrap_inline71 . The quadratic formula gives

displaymath73

which yields x= -1 or x=2. Therefore, the expression tex2html_wrap_inline79 is negative or equal to 0 when tex2html_wrap_inline81 .

 

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

 

Consider the series tex2html_wrap_inline197 and its associated sequence of partial sums tex2html_wrap_inline199 . We will say that tex2html_wrap_inline197 is convergent if and only if the sequence tex2html_wrap_inline199 is convergent. The total sum of the series is the limit of the sequence tex2html_wrap_inline199 , which we will denote by

displaymath207

 

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 tex2html_wrap_inline199 with the convergence of the sequence of numbers tex2html_wrap_inline211 . 

Basic Properties.

1.
Consider the series tex2html_wrap_inline197 and its associated sequence of partial sums tex2html_wrap_inline199 . Then we have the formula

displaymath217

for any tex2html_wrap_inline219 .
This implies in particular that if we know sequence of partial sums tex2html_wrap_inline199 , one may generate the numbers tex2html_wrap_inline223 since we have

displaymath225

 

2.
If the series tex2html_wrap_inline197 is convergent, then we must have

displaymath229

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

displaymath231

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

displaymath235

 

 

4.
(Algebraic Properties of convergent series) Let tex2html_wrap_inline197 and tex2html_wrap_inline239 be two convergent series. Let tex2html_wrap_inline241 and tex2html_wrap_inline243 be two real numbers. Then the new series

displaymath245

is convergent and moreover we have

displaymath247

 

 

Example. Show that the series

egin{displaymath}sum_{n geq 1} ln left(rac{n+1}{n}ight)end{displaymath}

is divergent, even though

displaymath251

 

Answer. Note that for any tex2html_wrap_inline219 , we have

displaymath255

Hence we have (for the associated partial sums)

displaymath257

Since tex2html_wrap_inline259 , then we have

displaymath261

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

displaymath251

since

displaymath265

which implies

displaymath267

 

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

displaymath269

 

Answer. We have

displaymath271

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

displaymath273

which gives

displaymath275

 

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

displaymath277

 

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

displaymath279

 

We recognize a geometric series. Since tex2html_wrap_inline281 , then the series is convergent and we have

displaymath283

 

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.

 

 

 

 

 

 


Comments (3)


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 !!!!!!!!



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