Quadratic Expressions & Equations

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GMAT 2009

Quadratic Expressions & Equations

Author: Sureshbala

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 $a\alpha^2 + b\alpha + c = 0$ then $\alpha$ 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

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

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

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

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 AND

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

If the equation ax ^2 + bx + c = 0 has equal roots then and 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

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

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

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

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

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

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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26 Comments

bala_ch said – Thu, 11 Sep 2008 17:00:31 -0000 ( Flag Edit 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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gauravjain26 said – Mon, 22 Sep 2008 15:54:54 -0000 ( Flag Edit Link )

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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1. Sureshbala said – Tue, 23 Sep 2008 06:03:43 -0000 ( Flag Edit Link )
  
  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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ajit2767 said – Fri, 26 Sep 2008 08:04:10 -0000 ( Flag Edit Link )

THIS LESSON IS EXTREMELY GOOD. GOOD WORK.
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Dhamodharan2008 said – Mon, 10 Nov 2008 08:05:31 -0000 ( Flag Edit Link )

Thanks For Useful tips
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pratim said – Thu, 20 Nov 2008 10:10:17 -0000 ( Flag Edit Link )

this is a extremely good lesson
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Riyana said – Sun, 23 Nov 2008 10:02:35 -0000 ( Flag Edit Link )

Great contribution. Thanks a lot.
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sriraam said – Fri, 28 Nov 2008 13:39:06 -0000 ( Flag Edit Link )

nice work budddy
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ashokle4 said – Tue, 02 Dec 2008 19:20:59 -0000 ( Flag Edit 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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1. pooja dubey said – Thu, 03 Sep 2009 14:56:41 -0000 ( Flag Edit Link )
  
  hi… here it mean that discriminant should be perfect square
  
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Sureshbala said – Thu, 04 Dec 2008 20:46:37 -0000 ( Flag Edit Link )

The discriminant must be a perfect square
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shivaprasad said – Sat, 13 Dec 2008 07:37:34 -0000 ( Flag Edit Link )

great
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ronak patel said – Tue, 16 Dec 2008 06:06:12 -0000 ( Flag Edit Link )

you are given a best formula provided to all
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shashishankar_thakur said – Tue, 30 Dec 2008 12:22:17 -0000 ( Flag Edit Link )

It is very good learning
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mainu_kalita said – Thu, 08 Jan 2009 11:16:17 -0000 ( Flag Edit Link )

Thankz :) Hey buddy keep up the good work…
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arvindsolanki said – Sat, 17 Jan 2009 17:17:56 -0000 ( Flag Edit Link )

thanks a lot sir…......... its very beneficial for us really thanks again
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sonaligrover1 said – Tue, 10 Feb 2009 11:48:07 -0000 ( Flag Edit Link )

thanx for the lesson..
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tsandeep said – Wed, 04 Mar 2009 13:03:50 -0000 ( Flag Edit 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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1. ashiwiniarya said – Sun, 15 Nov 2009 18:06:21 -0000 ( Flag Edit Link )
  
  yaaaaaaaaaaaaa….........the basic concept is eliminated
  
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deepakkumar753 said – Thu, 19 Mar 2009 11:49:28 -0000 ( Flag Edit Link )

very nice thanks….................
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gargi_l said – Mon, 06 Apr 2009 14:12:29 -0000 ( Flag Edit 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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anita yadav said – Tue, 14 Apr 2009 20:00:52 -0000 ( Flag Edit Link )

sir what is the meaning of TEX-ERROR????? i m nt getting this .pls explain me.
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1. oLahav said – Sun, 30 Aug 2009 14:51:36 -0000 ( Flag Edit Link )
  
  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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POORNIMA RAMESH said – Fri, 17 Apr 2009 14:23:29 -0000 ( Flag Edit 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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Sureshbala said – Wed, 06 May 2009 11:12:29 -0000 ( Flag Edit 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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lamaster said – Sun, 09 Aug 2009 14:50:11 -0000 ( Flag Edit Link )

thanx.
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pooja dubey said – Thu, 03 Sep 2009 14:57:38 -0000 ( Flag Edit Link )

sir can u please explain the very first line .
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POOJACHAUDHARY said – Fri, 18 Sep 2009 07:56:05 -0000 ( Flag Edit Link )

thanxx a lot sir its vry comfortable lesson really thaxxx sir POOJA CHAUDHARY
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sherwood said – Tue, 29 Sep 2009 16:41:08 -0000 ( Flag Edit Link )

very good mathematics specially quadratic aap hamesha mistakes ko hi q dekhte hai
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dedlee said – Tue, 06 Oct 2009 08:42:06 -0000 ( Flag Edit Link )

i think its poorly edited
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