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Quadratic Expressions & Equations

by Suresh


QUADRATIC EXPRESSION: If a \not=0, b, c are complex numbers then  ax^2 + bx +c is called a quadratic expression in x.


QUADRATIC EQUATION: If a \not=0,b,c are complex numbers then  ax^2 + bx + c = 0 is called a quadratic equation in x.


ROOT OF A QUADRATIC EQUATION: If a\alpha^2 + b\alpha + c = 0then \alpha is a root or solution of the quadratic equation ax^2 + bx + c = 0.

A quadratic equation can not have more than two roots or two solutions. The roots of ax^2 + bx + c = 0 are \frac{-b\pm \sqrt{b^2 - 4ac}}{2a} and its discrminent is \triangle = b^2 -4ac.


NATURE OF THE ROOTS OF THE EQUATION ax^2 + bx + c = 0

  1. If a,b,c are real and \triangle>0, then the roots are real and distinct.
  2. If a,b,c are real and \triangle=0, then the roots are real and equal.
  3. If a,b,c are real and \triangle<0, then the roots are two conjugate complex numbers.
  4. If a,b,c are rational and \triangle>0, and is a perfect square then the roots are rational and distinct.
  5. If a,b,c are rational and \triangle>0, and is not a perfect square then the roots are conjugate surds i.e \alpha\pm \beta.
  6. If a,b,c are rational and \triangle<0, then the roots are conjugate complex numbers i.e, \alpha\pm i\beta.

FORMATION OF THE QUADRATIC EQUATION WITH ROOTS \alpha AND \beta:
The quadratic equation whose roots are \alpha and \beta is
x^2 - (\alpha + \beta)x + \alpha \beta = 0 \Rightarrow (x-\alpha)(x-\beta) = 0 .


RELATION BETWEEN THE ROOTS \alpha , \beta OF ax^2 + bx + c = 0.

1.  \alpha + \beta = \frac{-b}{a} , \alpha \beta = \frac{c}{a}.

2. |\alpha - \beta| = \frac {\sqrt{b ^2 - 4ac}}{|a|}.

3.  \alpha ^2 + \beta ^2 = \frac{b ^2-2ac}{a ^2}.

4.  \alpha ^3 + \beta ^3 = \frac{3abc-b ^3}{a ^3}.

5.  \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-b}{c}.

6.  \frac{1}{\alpha ^2} + \frac{1}{\beta ^2} = \frac{b ^2 -2ac}{c ^2}.

7.  \frac{1}{(a\alpha + b)} + \frac{1}{(a\beta+b)} = \frac{b}{ac}.

8.  \frac{1}{(a\alpha + b) ^2} + \frac{1}{(a\beta +b) ^2} = \frac{ b ^2 -2ac}{a ^2 c ^2}.

9.  \frac{1}{(a\alpha + b) ^3} + \frac{1}{(a\beta +b) ^3} = \frac{ b ^3 -3abc}{a ^3 c ^3}.

10.  |\alpha ^2 - \beta ^2| = \frac{|b|\sqrt{b ^2 -4ac}}{a ^2}.


PROPERTIES OF ROOTS OF THE EQUATION  ax^2 + bx +c = 0.

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

If a and c are of opposite sign i.e, \frac{c}{a} 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  \frac{c}{a}.

If a+c=b, then the roots are -1 and  \frac{-c}{a}.

If the roots are in the ratio m:n then  (m+n) ^2 ac = mnb ^2.

If one root is p times the other root then  (1+p) ^2 ac = pb ^2.

If one root is equal to the n th power of the other root then
 (ac^n) ^\frac{1}{n+1} + (a ^nc) ^\frac{1}{n+1} +b = 0.

If one root is square of the other, then  a ^2 c + ac ^2 = b(3ac-b ^2).

If roots differ by unity, then  b ^2 = 4ac + a ^2.


SAME ROOTS: If  a_1 x ^2 + b_1 x + c_1 = 0 and a_2 x^2 + b_2 x + c_2 = 0 have the same roots then  \frac{a_1}{a_2} = \frac {b_1}{b_2} = \frac{c_1}{c_2}.


ONE ROOT IS COMMON: The equations  a_1 x ^2 + b_1 x + c_1 + 0 and  a_2 x ^2 + b_2 x + c_2 = 0 where  a_1 b_2 - a_2 b_1 \not=0, a_1,a_2 \not=0, have one common root then  (c_1 a_2 - c_2 a_1) ^2 = (a_1 b_2 - a_2 b_1)(b_1 c_2 - b_2 c_1) and the common root is
 \frac {c_1 a_2 - c_2 a_1}{a_1b_2 - a_2 b_1}.


SIGNS OF 'a' AND  ax ^2 + bx + c:

If the equation  ax ^2 + bx + c = 0 has complex roots (\triangle < 0) then  a and  ax ^2 + bx + c will have the same sign \forall x \in R

If the equation  ax ^2 + bx + c = 0 has equal roots then  a and  ax ^2 + bx + c will have same sign \forall x \in R - [\frac{-b}{2a}]

If the equation  ax ^2 + bx + c = 0 has real roots  \alpha , \beta (\triangle > 0, \alpha < \beta) then

1. \alpha < x < \beta \Leftrightarrow a and  ax ^2 + bx + c will have opposite sign.

2. x<\alpha or  x > \beta \Leftrightarrow a and  ax ^2 + bx + c will have same sign.


MAXIMUM OR MINIMUM VALUE O QUADRATIC EXPRESSION

If a > 0, then the minimum value of  ax ^2 + bx + c is  \frac{4ac-b ^2}{4a} (This value is attained at  x = \frac{-b}{2a} ).

If a < 0, then the maximum value of  ax ^2 + bx + c is  \frac{4ac-b ^2}{4a} (This value is attained at  x = \frac{-b}{2a} ).
________________________________________________________________________________________

If (\alpha < \beta), (x-\alpha)(x-\beta) \underline{<} 0 \Leftrightarrow x \in [\alpha, \beta] \quad i.e, \quad \alpha \underline{<} x \underline{<} \beta.


If (\alpha < \beta), (x-\alpha)(x-\beta) \underline{>}0 \Leftrightarrow x \in [-\infty, \alpha] \cup [\beta, \infty] i.e, x \underline{ <} \alpha or x \underline{>} \beta.


If \alpha, \beta are the roots of  f(x)=ax ^2 + bx + c = 0 then the equation whose roots are

1.  -\alpha, -\beta \quad is \quad f(-x) = 0.

2.  \frac{1}{\alpha}, \frac{1}{\beta} \quad is \quad f(\frac{1}{x}) = 0.

3.  k\alpha, k\beta \quad(k \not=0) \quad is \quad f(\frac{x}{k}) = 0.

4.  \alpha + k, \beta + k \quad is \quad f(x-k) = 0.

5.  \alpha ^2, \beta ^2 \quad is \quad f(\sqrt{x}) = 0.

6.  \alpha ^k, \beta ^k \quad is \quad f(\sqrt[k]{x}) = 0.

7.  \frac{\alpha}{1+\alpha}, \frac{\beta}{1+\beta} \quad is \quad f(\frac{x}{1-x}) = 0.


LOCATING THE ROOTS OF QUADRATIC EQUATION UNDER GIVEN CONDITIONS

Both the roots of equation  ax^2 + bx + c = 0 are greater than a given number 'k' if

  1. \quad \triangle \underline > 0.
  2. \quad af(k) > 0.
  3. \quad\frac{-b}{2a} > k.

Both the roots of equation  ax^2 + bx + c = 0 are smaller than a given number 'k' if

  1. \quad \triangle \underline > 0.
  2. \quad af(k) > 0.
  3. \quad\frac{-b}{2a} < k.

Exactly one root of  ax^2 + bx + c = 0 lies between the numbers  'p' \quad and \quad 'q' if

f(p)f(q)\underline<0

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

Both the roots of equation  ax^2 + bx + c = 0 lie between two given numbers 'p'\quad and \quad 'q' \quad (p<q) if

  1. \quad \triangle \underline > 0.
  2. \quad af(p) > 0.
  3. \quad af(q) > 0.
  4. \quad p < \frac{-b}{2a} < q.

The extreme values of  f(x) = \frac{x ^2 - ax +b}{x ^2 + ax +b} are  f(\sqrt{-b}), f(\sqrt{b}).
________________________________________________________________________________________

28 Comments
    shashicse
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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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    satyaprakashsingh1986
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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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    ragaswanth
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    ragaswanthMon, 10 May 2010 07:36:44 -0000

    nice for learning fundamentals

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    kg00710
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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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    dedlee
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    DEEPANSHU GUPTATue, 06 Oct 2009 08:42:06 -0000

    i think its poorly edited

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

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    Sureshbala
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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 RAMESH
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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 yadav
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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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    oLahav
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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_l
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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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    deepakkumar753
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    deepak agarwalThu, 19 Mar 2009 11:49:28 -0000
    very nice

    thanks………………..

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

    yaaaaaaaaaaaaa…………the basic concept is eliminated

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    sonaligrover1
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    sonaligrover1Tue, 10 Feb 2009 11:48:07 -0000

    thanx for the lesson..

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

    It is very good learning

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    ronak patel
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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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    shivaprasad
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    shivaprasadSat, 13 Dec 2008 07:37:34 -0000

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

    The discriminant must be a perfect square

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    ashokle4
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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 dubey
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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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    sriraam
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    sriraamFri, 28 Nov 2008 13:39:06 -0000

    nice work budddy

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

    Great contribution. Thanks a lot.

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

    this is a extremely good lesson

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    Dhamodharan2008
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    DhamodharanMon, 10 Nov 2008 08:05:31 -0000

    Thanks For Useful tips

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    ajit2767
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    ajit2767Fri, 26 Sep 2008 08:04:10 -0000

    THIS LESSON IS EXTREMELY GOOD. GOOD WORK.

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    gauravjain26
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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 ax^2 + bx + c = 0 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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    Sureshbala
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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 x^2i.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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    bala_ch
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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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