01. quadratic equations-1

8/10/2019 01. Quadratic Equations-1

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Quadratic Equation- Session1


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Session Objective

1. Definition of important terms(equation,expression,pol nomial,identit ,quadratic etc.!

". #indin$ roots b factori%ation met&od

'. eneral solution of roots .

). *ature of roots


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Quadratic Equation - Definitions(Expression + Equation!

Equation Statement of equalitbet een t o expression

Expression

/epresentation of relations&ipbet een t o (or more! variables

0 ax " 2bx2c,

ax " 2 bx 2 c

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/oot -value(s! for &ic& a equation satisfies

x " -)x2' (x-'!(x-1!

x ' or 1 satisfies x " -)x2'

/oots of x " -)x2'

Example


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Quadratic EquationDefinitions (5ol nomial!

5ol nomial

5(x! a 2 a 1x 2 a " x " 2 6 2 a nx n ,

7 pol nomial equation of de$ree n al a s &ave n roots

/eal or non-real

&i$&est po er of t&e variable

&ere a , a1, a

", 6 a

nare coefficients ,

and n is positive inte$erDe$ree of t&e pol nomial

34 1

( )na


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Equation " roots (sa 1,"!(x-1!(x-"! x " - 'x2"

" nd de$ree equation " roots

" nd de$reeequation

8 /oots are 1 ,"(x- 1 !(x- " ! x " -( 12 " !x2 1"

ax " 2 bx 2 c

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8 /oots are 1 ," ,'(x- 1 !(x- " ! (x- ' !

ax ' 2bx " 2cx2d ' rd de$ree equation

' rd de$ree equation ' roots

8 /oots are 1 ," , ' ,66. n(x- 1 !(x- " ! (x- ' !6.. (x- n!

a nx n2a n-1 x n-1 266.2 a n t& de$ree equation

n t& de$ree equation n roots

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Quadratic EquationDefinitions (Quadratic + /oots!

Quadratic 7 pol nomial of de$ree "

7 quadratic equation al a s &as t o roots

ax " 2bx2c

ax " 2bx2c is a quadratic equation . (a !

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/oots

x -a 9:&ere is t&e " nd root of quadratic

equation9

;&en &at is itsdifference from

x2a

(x2a! "

(x2a!(x2a!

x -a, -a

t o roots7lso satisfies condition forquadratic equation

:&at are t&e roots of t&e equation

(x2a!"

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5ol nomial identit


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5ol nomial identit

e terms is same on bot& t&e side


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Quadratic Equation-#actori%ation Aet&od

Solve for x " 2x-1"

Step1 product-1"

-),'-",B

),-'

Step"

factorsSum of factors

-1

1

)

Step' x " 2()-'!x -1"

(x2)!(x-'!/oots are -), '

factors it& opposite si$n

34 "

x " 2)x-'x-1"


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Quadratic Equation-#actori%ation Aet&od

x " 2x-1" x " 2()-'!x -1"( &ere roots are C),'!

Similarl if ax " 2bx2c &as roots ,ax " 2bx2c a(x " -( 2 !x 2 !

=omparin$ co-efficient of li>e terms a a( ! aa b c

+ = =b( ! sum of t&e rootsa

+ = = c

5r oduct of t&e rootsa

= =

34 "


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5roperties of /oots

Quadratic equation ax " 2bx2c ,a,b,c / and

;&e equation becomes a x " 2 (b a!x 2 (c a! F

a {x " -( 2 !x2 } a(x- !(x- !

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x " -(sum! x2(product!


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Step 1 equation of roots C1G + -)

(x21G!(x2)!

Step" et t&e ori$inal equation

x " 21Bx2B

Or x " 21Hx2B

/oots are C1 + -B


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5roduct of t&e roots of t&e equationx " 2Bx2 " 21 is C",;&en t&evalue of is(a!-", (b!-1, (c!", (d!1

ID=E-1HHHJ


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5roduct of t&e roots ( " 21! -"

(21! " -1

x " 2Bx2 " 21

5roduct of t&e roots of t&e equation

x " 2Bx2 " 21 is C",;&en t&e value of is(a!-", (b!-1, (c!", (d!1

34 G


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eneral Solution

;o find roots of ax"

2 bx 2 c

Step 1

=onvert it in perfect square term

i. Aultipl in$ t&is equation b )a,

)a " x " 2 )abx 2 )ac

ii. 7dd and subtract b "

()a " x " 2 )abx 2 b " ! 2 )ac - b "

4O: KK

("ax 2 b! " b " - )ac

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ac4bbax2 2 =+

ax " 2 bx 2 c &as t o roots as

a2ac4bb

;a2ac4bb 22

=+

=

Step " Solve #or x

"-b L b - )acx

"a

34 'eneral Solution


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eneral Solution

(b"

- )ac! discriminant of t&equadratic equation, and is denotedb D .

a2

Db;

a2

Db =

+=

/oots are

;&is is called t&e $eneral solution of a quadratic

equation

34 '


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#ind t&e roots of t&e equation

x " -1 x2""

4ere a 1, b -1 , c ""

7ppl t&e $eneral solution form( 1 ! 1 ).1."" 1 " '

".1 " + +

= =

( 1 ! 1 ).1."" 1 " '

".1 "

= =

34 '


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*ature of /oots

Discriminant, D b"

-)ac

a"Db

, =

/oots are real

D /oots are real and equal

D N is not real D /oots are ima$inar

(D is not a perfect square!


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#ind t&e nature of t&e roots of t&eequation

x " 2"('a2G!x2"(Ha " 2"G!

D )('a2G! " -)."(Ha " 2G! -'Ba " 21" a-1

-)('a-G!"

DN

/oots are ima$inar except a G '

7s ('a-G!"

except a G '

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=omplex /oots Occur in 5air


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_H004


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()->!x " 2(">2)!x2(P>21! becomes

a perfect square

(a! ' or (b! ) or

(c ! ' or ) (d! *one of t&ese

34 )

4int ()->!x " 2(">2)!x2(P>21! becomes

a perfect square

/oots of t&e correspondin$ equation are equal


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!x " 2(">2)!x2(P>21! &as equal roots

D (">2)! " -).()->!.(P>21!

)> " 21B>21B-)('1>-P> " 2)!

> " 2)>2)2P> " -'1>-)

H>" -"R> > or '

34 )#or &at values of >

()->!x " 2(">2)!x2(P>21! becomes

a perfect square

(a! ' or (b! ) or

(c ! ' or ) (d! *one of t&ese


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=lass Exercise1

*umber of roots of t&e equation

(x 2 1!'

C (x C 1!'

are(a! t o (b! t&ree (c! four(d! *one of t&ese

Solution (x 2 1! ' C (x C1! '

Bx " 2"

"('x " 21! ,


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=lass Exercise"

(x 2 1! ' " x ' 2( 2"! x " 2 (a C " !x2 b is an identit , ;&en value of( , a, b! are(a! (-1, G, 1!(b! ( 1,G,1! (c! (1,G ,1! (d! *one of t&ese

Solution

Since (x 2 1! ' " x ' 2 ( 2"! x " 2(a C"!x 2 b is anidentit , co-efficient of li>e terms of bot& t&e sidesare t&e same

x ' 2 'x " 2 'x 2 1 " x ' 2 ( 2"! x " 2(a C"!x 2 b

" 1-------(i!2" '---(ii!


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=lass Exercise"

(x 2 1! ' " x ' 2( 2"! x " 2 (a C " !x 2 b is

an identit , ;&en value of ( , a, b! are(a! (-1, G, 1! (b! ( 1,G,1! (c! (1, G ,1!(d! *one of t&ese

" 1-------(i!

2" '---(ii!

aC" ' a Gb 1

1


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=lass Exercise'

/oots of t&e equation cx"

C cx 2 c 2bx " C cx C b are(a! c and b (b!1 ,(c! (c 2 b! and (c C b! (d! *one of

t&ese

Solution (c 2 b!x " C "cx 2 (c C b!

(c2b!x " C (c2b!2(cCb!Fx2(cCb!

(c2b!x " C(c2b!xC(cCb!x2(c Cb!

(c2b!x (x C 1! C (c C b! (x C 1!

(x C 1! (c2b!x C(c C b!F

/oots are 1 and

c bc b

+


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=lass Exercise)

(x-a!(x-b! c , are t&e roots

x"

-(a2b!x2 ab-cSo 2 (a2b!M ab-c66(1!

*o (x- !(x- !2c x " -( 2 !x2 2cx " -(a2b !x2 ab b (1!

(x-a! (x-b! /oots are a and b

?et , are t&e roots of t&e equation (x-a!(x-b! c, c . ;&en roots of t&e equation (x- !(x- !2c are(a! a,c (b!b,c (c ! a,b (d!(a2c!,(b2c!


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=lass ExerciseG

G.;&e equation &ic& &as G2

' and)2 " as t&e onl roots is(a!never possible(b! a quadratic equation it&rational co-efficient

(c! a quadratic equation it&irrational co-efficient(d! not a quadratic equation

Since it &as t o roots it is a quadratic equation.

Solution

7s irrational roots are not in conju$ate form. =o-efficientare not rational.


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=lass ExerciseB


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=lass ExerciseB


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=lass ExerciseR

@ot& t&e roots of t&e equation(xCb!(xCc!2(xCc!(xCa!2(xCb!(xCa!are al a s a,b,c, /(a! Equal (b!


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=lass ExerciseR

@ot& t&e roots of t&e equation (x C b! (x C

c! 2 (x C c! (x C a! 2 (x C b! (x C a! areal a s a,b,c, /(a! Equal (b!


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=lass ExerciseP

;&e roots of t&e equation(a2b2c!x " C"(a2b!x2(a2bCc! are($iven t&at a, b, c are rational.!(a! /eal and equal (b! /ational

(c!


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=lass Exercise H

Solution D

) (ac2bd! " - ) (a " 2b " !(c " 2d " !

a " c " 2 b " d " 2 "abcd (a " 2 b " ! (c " 2 d " !

"abcd a " d " 2 b " c " a " d " 2 b " c " C "abcd

(ad C bc! "

ad C bc ad bca cb d

=


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=lass Exercise1


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=lass Exercise1


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=lass Exercise1

01. quadratic equations-1

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