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Quadratic Expressions & Equations

by Suresh

QUADRATIC EXPRESSION: If are complex numbers then is called a quadratic expression in x.

QUADRATIC EQUATION: If are complex numbers then is called a quadratic equation in x.

ROOT OF A QUADRATIC EQUATION: If then is a root or solution of the quadratic equation

A quadratic equation can not have more than two roots or two solutions. The roots of are and its discrminent is .

NATURE OF THE ROOTS OF THE EQUATION

1. If a,b,c are real and , then the roots are real and distinct.
2. If a,b,c are real and , then the roots are real and equal.
3. If a,b,c are real and , then the roots are two conjugate complex numbers.
4. If a,b,c are rational and , and is a perfect square then the roots are rational and distinct.
5. If a,b,c are rational and , and is not a perfect square then the roots are conjugate surds i.e .
6. If a,b,c are rational and , then the roots are conjugate complex numbers i.e, .

FORMATION OF THE QUADRATIC EQUATION WITH ROOTS AND :
The quadratic equation whose roots are and is
.

RELATION BETWEEN THE ROOTS OF

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

PROPERTIES OF ROOTS OF THE EQUATION

If a and c are of the same sign i.e, is +ve, then both the roots are of same sign.

If a and c are of opposite sign i.e, is -ve, then the roots are of opposite sign.

If both the roots are -ve, then a,b,c will have the same sign.

If both the roots are +ve then a, c will have the same sign different from the sign of b.

If a=c, then the roots are reciprocal to each other.

If a+b+c=0,then the roots are 1 and

If a+c=b, then the roots are -1 and

If the roots are in the ratio m:n then

If one root is p times the other root then

If one root is equal to the n th power of the other root then

If one root is square of the other, then

If roots differ by unity, then

SAME ROOTS: If and have the same roots then

ONE ROOT IS COMMON: The equations and where , have one common root then and the common root is

SIGNS OF AND

If the equation has complex roots then and will have the same sign

If the equation has equal roots then and will have same sign

If the equation has real roots then

1. and will have opposite sign.

2. or and will have same sign.

MAXIMUM OR MINIMUM VALUE O QUADRATIC EXPRESSION

If a > 0, then the minimum value of is (This value is attained at ).

If a < 0, then the maximum value of is (This value is attained at ).
________________________________________________________________________________________

If are the roots of then the equation whose roots are

1.

2.

3.

4.

5.

6.

7.

LOCATING THE ROOTS OF QUADRATIC EQUATION UNDER GIVEN CONDITIONS

Both the roots of equation are greater than a given number if

Both the roots of equation are smaller than a given number if

Exactly one root of lies between the numbers if

but f(p) and f(q) are not simultaneously zero.

Both the roots of equation lie between two given numbers if

The extreme values of are
________________________________________________________________________________________

29 Comments
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samseeSun, 18 Jan 2015 18:30:24 -0000

Differentiate quadratic equation and quadratic expression

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shashicseSat, 27 Jul 2013 14:19:53 -0000

ye kaun sa language use kia hai vai,,headache ho gaya

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satyaprakashsingh1986Mon, 08 Aug 2011 05:11:20 -0000

Hi
Please quote the reference.
I am weak in maths so I am going step by step.
Please explain me in detail the first line of this article.
That is please restate it in some more detail.

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ragaswanthMon, 10 May 2010 07:36:44 -0000

nice for learning fundamentals

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krishan gopal agrawalMon, 01 Feb 2010 17:06:30 -0000

guys solve this…

100-100
_________=2
100-100

solve only L.H.S….. and prove that L.H.S.=R.H.S…………

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DEEPANSHU GUPTATue, 06 Oct 2009 08:42:06 -0000

i think its poorly edited

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anandsharmaTue, 29 Sep 2009 16:41:08 -0000

very good mathematics specially quadratic aap hamesha mistakes ko hi q dekhte hai

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pooja dubeyThu, 03 Sep 2009 14:57:38 -0000

sir can u please explain the very first line .

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master laSun, 09 Aug 2009 14:50:11 -0000

thanx.

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SureshWed, 06 May 2009 11:12:29 -0000

Hi Ramesh,

Don't worry. They mean one the same i.e both the roots will be complex conjugate numbers.

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POORNIMA RAMESHFri, 17 Apr 2009 14:23:29 -0000

hello sir,
i have a doubt
If a,b,c are real and discriminant is<0, then the roots are two conjugate complex numbers.
If a,b,c are rational and discriminant is<0, then the roots are conjugate complex numbers
here wat is the difference between the roots are two conjugate complex nos and the roots are conjugate complex nos?
doent they mean the same that the roots are complex conjugate nos?

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anita yadavTue, 14 Apr 2009 20:00:52 -0000

sir
what is the meaning of TEX-ERROR? i m nt getting this .pls explain me.

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Oren LahavTue, 14 Apr 2009 20:12:29 -0000

Tex Error means there's an error in the formula editor we use (it's call Tex). It's not a mathematical expression you need to be familiar with for any particular exam, don't worry.

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gargi_lMon, 06 Apr 2009 14:12:29 -0000

do u have a solution key to the lessons…i got some of the questions very slow…and few not at all…

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deepak agarwalThu, 19 Mar 2009 11:49:28 -0000
very nice

thanks………………..

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tsandeepWed, 04 Mar 2009 13:03:50 -0000

A MISTAKE …….

suresh bala … its the first time that i have come across quadratic equation that is defined for complex coefficients. As far as i remember … coefficents a,b & c needs to be real and a should not be equal to zero. Only then is that equation is consider as quadratic and not otherwise as you have claimed where a,b & c are complex numbers. But anyways you dont have to know this to prepare for CAT, as they dont wander about complex numbers i suppose.

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ashiwini kumar aryaSun, 15 Nov 2009 18:06:20 -0000

yaaaaaaaaaaaaa…………the basic concept is eliminated

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sonaligrover1Tue, 10 Feb 2009 11:48:07 -0000

thanx for the lesson..

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Arvind Singh SolankiSat, 17 Jan 2009 17:17:56 -0000

thanks a lot sir………… its very beneficial for us really
thanks again

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shashishankar_thakurTue, 30 Dec 2008 12:22:17 -0000

It is very good learning

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ronak patelTue, 16 Dec 2008 06:06:12 -0000

you are given a best formula provided to all

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shivaprasadSat, 13 Dec 2008 07:37:34 -0000

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SureshThu, 04 Dec 2008 20:46:37 -0000

The discriminant must be a perfect square

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ASHOK chakravarthy DADDALATue, 02 Dec 2008 19:20:59 -0000

sir, in nature of roots of ax2+bx+c=0. u mentioned that "If a,b,c are rational and discrminent > 0, and is a perfect square then the roots are rational and distinct." pls can you clarify that in this statement which is perfect square ?

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pooja dubeyThu, 03 Sep 2009 14:56:40 -0000

hi…
here it mean that discriminant should be perfect square

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sriraamFri, 28 Nov 2008 13:39:06 -0000

nice work budddy

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Sun, 23 Nov 2008 10:02:35 -0000

Great contribution. Thanks a lot.

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pratimThu, 20 Nov 2008 10:10:17 -0000

this is a extremely good lesson

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DhamodharanMon, 10 Nov 2008 08:05:31 -0000

Thanks For Useful tips

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ajit2767Fri, 26 Sep 2008 08:04:10 -0000

THIS LESSON IS EXTREMELY GOOD. GOOD WORK.

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Gaurav JainMon, 22 Sep 2008 15:54:54 -0000

The very fist line of the lesson mention that if b and c are complex numbers, then Only that is not true. Complex numbers are in form (a + bi)

I think there is some mistake, so please make the required corrections for readers' benefit.

regards

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SureshTue, 23 Sep 2008 06:03:43 -0000

hi gaurav……I think there's nothing wrong in my definition. In the above quadratic equation, what I meant is that the coefficient of i.e a is also complex numbers. I think you were mistaken that b and c are only complex. I clearly said that a (which is non zero)as well as b and c are all complex numbers. I hope this is clear.

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BalaSaraswathiThu, 11 Sep 2008 17:00:31 -0000

Thanks for a very good lesson and it's nice to have practice exercises at the end.
Can you please provide us some more questions to practice?

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Last Updated At Dec 07, 2012
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