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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
34 1
/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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Quadratic EquationDefinitions (5ol nomial!
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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Quadratic EquationDefinitions (5ol nomial!
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$inal equation
x " 21Bx2B
Or x " 21Hx2B
/oots are C1 + -B
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5roduct of t&e roots of t&e equationx " 2Bx2 " 21 is C",;&en t&evalue of is(a!-", (b!-1, (c!", (d!1
ID=E-1HHHJ
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5roduct of t&e roots ( " 21! -"
(21! " -1
x " 2Bx2 " 21
5roduct of t&e roots of t&e equation
x " 2Bx2 " 21 is C",;&en t&e value of is(a!-", (b!-1, (c!", (d!1
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eneral Solution
;o find roots of ax"
2 bx 2 c
Step 1
=onvert it in perfect square term
i. Aultipl in$ 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
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#ind t&e roots of t&e equation
x " -1 x2""
4ere a 1, b -1 , c ""
7ppl t&e $eneral solution form( 1 ! 1 ).1."" 1 " '
".1 " + +
= =
( 1 ! 1 ).1."" 1 " '
".1 "
= =
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*ature of /oots
Discriminant, D b"
-)ac
a"Db
, =
/oots are real
D /oots are real and equal
D N is not real D /oots are ima$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$inar except a G '
7s ('a-G!"
except a G '
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=omplex /oots Occur in 5air
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_H004
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()->!x " 2(">2)!x2(P>21! becomes
a perfect square
(a! ' or (b! ) or
(c ! ' or ) (d! *one of t&ese
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4int ()->!x " 2(">2)!x2(P>21! becomes
a perfect square
/oots of t&e correspondin$ 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