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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}).

Now let us get into some practice exercises

Basic Level Practice Exercise

Moderate Level Practice Exercise

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Ask The Experts


  1. bala_ch saidThu, 11 Sep 2008 17:00:31 -0000 ( Link )

    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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  2. gauravjain26 saidMon, 22 Sep 2008 15:54:54 -0000 ( Link )

    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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  3. ajit2767 saidFri, 26 Sep 2008 08:04:10 -0000 ( Link )

    THIS LESSON IS EXTREMELY GOOD. GOOD WORK.

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  4. Dhamodharan2008 saidMon, 10 Nov 2008 08:05:31 -0000 ( Link )

    Thanks For Useful tips

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  5. pratim saidThu, 20 Nov 2008 10:10:17 -0000 ( Link )

    this is a extremely good lesson

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  6. Riyana saidSun, 23 Nov 2008 10:02:35 -0000 ( Link )

    Great contribution. Thanks a lot.

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  7. sriraam saidFri, 28 Nov 2008 13:39:06 -0000 ( Link )

    nice work budddy

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  8. ashokle4 saidTue, 02 Dec 2008 19:20:59 -0000 ( Link )

    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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  9. Sureshbala saidThu, 04 Dec 2008 20:46:37 -0000 ( Link )

    The discriminant must be a perfect square

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  10. shivaprasad saidSat, 13 Dec 2008 07:37:34 -0000 ( Link )

    great

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  11. ronak patel saidTue, 16 Dec 2008 06:06:12 -0000 ( Link )

    you are given a best formula provided to all

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  12. shashishankar_thakur saidTue, 30 Dec 2008 12:22:17 -0000 ( Link )

    It is very good learning

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  13. arvindsolanki saidSat, 17 Jan 2009 17:17:56 -0000 ( Link )

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

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  14. sonaligrover1 saidTue, 10 Feb 2009 11:48:07 -0000 ( Link )

    thanx for the lesson..

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  15. tsandeep saidWed, 04 Mar 2009 13:03:50 -0000 ( Link )

    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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  16. deepakkumar753 saidThu, 19 Mar 2009 11:49:28 -0000 ( Link )

    very nice

    thanks………………..

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  17. gargi_l saidMon, 06 Apr 2009 14:12:29 -0000 ( Link )

    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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  18. anita yadav saidTue, 14 Apr 2009 20:00:52 -0000 ( Link )

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

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  19. POORNIMA RAMESH saidFri, 17 Apr 2009 14:23:29 -0000 ( Link )

    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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  20. Sureshbala saidWed, 06 May 2009 11:12:29 -0000 ( Link )

    Hi Ramesh,

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

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  21. lamaster saidSun, 09 Aug 2009 14:50:11 -0000 ( Link )

    thanx.

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  22. pooja dubey saidThu, 03 Sep 2009 14:57:38 -0000 ( Link )

    sir can u please explain the very first line .

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  23. POOJACHAUDHARY saidFri, 18 Sep 2009 07:56:05 -0000 ( Link )

    thanxx a lot sir its vry comfortable lesson really thaxxx sir POOJA CHAUDHARY

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  24. sherwood saidTue, 29 Sep 2009 16:41:08 -0000 ( Link )

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

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  25. dedlee saidTue, 06 Oct 2009 08:42:06 -0000 ( Link )

    i think its poorly edited

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  26. kg00710 saidMon, 01 Feb 2010 17:06:30 -0000 ( Link )

    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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