03. quadratic equations-3

8/10/2019 03. Quadratic Equations-3

1/46

Quadratic EquationSession 3


2/46

Session Objective

1. Condition for common root

2. Set of solution of quadratic inequation

3. Cubic equation


3/46

Condition for Common Root

The equations ax2+ bx + c = 0 &

ax2

+ bx + c = 0 has a commonroot(CR)

a 2+ b + c = 0

a 2+ b + c = 0

By rule of cross-multiplication

Treating 2 and astwo different variable

2 1

bccb caac abba

_H007


4/46

Condition for Common Root

''

''

baab

cbbc

2

'ba'ab

'ac'ca

Condition for common root ofax2+ bx + c = 0 &ax2+ bx + c = 0 is

(ca-ac)2=(bc-cb)(ab-ba)

_H007


5/46

Illustrative Problem

If x2-ax-21=0 and x2-3ax+35=0(a>0)has a common root thenvalue of a is

(a)3 (b) 4 (c )2 (d) 4

_H007


6/46


Let be the common root

2-a-21=0 2-3a+35=0

2 1

35a 63a 35 ( 21) 3a ( a)

By Cross- Multiplication

2 1

98a 56 2a

Solution: Method 1

If x2-ax-21=0 and x2-3ax+35=0(a>0)has a common root then value of a

is(a)3 (b)4 (c )2 (d)4

_H007


7/46


2 1

98a 56 2a

2 2849 anda

a2=16 a = 4

As a>0 a=4

If x2-ax-21=0 and x2-3ax+35=0

(a>0)has a common root then value of ais(a)3 (b) 4 (c )2 (d) 4

228

49a

_H007


8/46


If x2-ax-21=0 and x2-3ax+35=0

(a>0)has a common root then value ofa is(a)3 (b) 4 (c )2 (d) 4

2- a- 21 = 0 .(A)

2-3a+35 = 0 ..(B)

(A) (B) 2a= 56 28

a

Substituting in (A)

228 28a 21 0

a a

228 49a

a = 4

As a>0 a=4

Solution: Method 2

_H007


9/46


If equation x2-ax+b=0 and

x2

+bx-a=0 has only one commonroot then prove that a-b=1

Solution:

x2- ax + b=0 (A)

x2 + bx - a=0 (B)

By observation at x=1 both the

equation gives same value.

L.H.S. = a-b-1 for x=1

This means x=1 is the common root

a b 1= 0 ab=1

Why?for x=1 both the

equations give this

_H007


10/46


If equation x2-ax+b=0 and

x2

+bx-a=0 has only one commonroot then prove that a-b=1

Solution:

Let be the common root then2 -a+ b = 0 &2 + b- a = 0 subtracting onefrom the other we get(b + a) - (b + a) = 0

= 1 provided b + a 0Hence x = 1 is the common root1 a + b = 0 or a b = 1

_H007

Method 2

Why??


11/46

Condition for Two CommonRoots

The equations ax2+bx+c=0 andax2+bx+c = 0 have both rootscommon

For two roots to be common

a b c

a' b ' c '

ax2+ bx + c K(ax2+ bx + c)

why?when both the roots arecommon ,two equations will be

same .

But not necessarily identical.

As x23x+2=0 and 2x26x+4=0

Same equation.Both have roots 1,2But not identical

_H007


12/46


If x2+ax+(a-2) = 0 and

bx

2

+2x+ 6 = 0 have both rootscommon then a : b is(a) 2 (b)1/2(c) 4 (d)1/4

Solution:

As both roots are common

1 a a 2

b 2 6

1 1b 2

b 2

a a 2

2 6

a 1

b 2

3a a 2

a=-1

_H007


13/46

Quadratic Inequation

If ax2+bx+c =0 has roots ,; let 0

ax2+bx+c 0

A statement of inequalityexist between L.H.S andR.H.S

QuadraticInequation

When ax2+bx+c >0 Let a>0

(x- )(x- )>0

_H009


14/46


(x-)(x-

)>0

Either(x- )>0; (x- )>0

Or(x- )>

x>x < x and x<

and arenot includedin set ofsolutions

-

_H009


15/46


Find x for which 6x2-5x+1>0

holds true

Either x>1/2 or x0

Solution:

_H009


16/46


When ax2+bx+c < 0

(x- )(x- ) < 0

ax2+bx+c = a(x- )(x- )

and a>0

Either (x- )0 Or (x- )>0; (x- )


17/46


1/3 1/2x

1/3


18/46


Solve for x : x2- x 6 > 0

(x-3) (x+2) >0

Step1:factorize into linear terms

-2 3

Step2 :Plot x for which x2-x6=0 on number line

As sign of a >0 x2-x6 >0 for either x3

-

Solution:

_H009


19/46


For a(x- )(x- ) 0

x

x x

Here set of solution contains ,and all values outside ,

For a(x- )(x- ) 0

x

x

x lies within ,and alsoincludes ,in solution set

_H009


20/46


2If x 8x + 10 0, then values of x are

(a) 4 6 x 4 6

(b) x 4 6 or x 4 6

(c) 8 6 x 8 6

(d) None of these

Solution : x28x + 10 0

step1: Find the roots of the corresponding equation

Roots of x28x + 10 = 0 are

, 4 6

_H009


21/46



(a) 4 6 x 4 6

(b) x 4 6 or x 4 6

(c) 8 6 x 8 6 (d) None of these

x 4+ 6 x4 6 0 x - 4 - 6 x - 4+ 6 0

x 4 6 or x 4 6

Step2: Plot on number-line

4-6 and 4+6are included inthe solution set

x28x + 10 0

4-6 4+6-

_H009


22/46


Solve for x : - x2+15 x 6 > 0

Here a=-1Solution:

Step1: Multiply the inequation

with (-1)to make a positive.

Note- Corresponding sign of inequality will also change

x215 x + 6 < 0

(x-7)(x-8)


23/46

Cubic Equation

P(x)=ax

3

+bx

2

+cx+d

A polynomial of degree 3

P(x)=0 ax3+bx2+cx+d=0 is a cubic equationwhen a 0

Number of roots of a cubicequation?

_H015


24/46

Cubic Equation

Let the roots of ax3

+bx2

+cx+d =0be ,,

ax3+bx2+cx+d

a(x- ) (x- ) (x- )

As ax2+bx+c has roots ,can be writtenas ax2+bx+c a(x- ) (x- )

a[x3-(++)x2+(++)x-()]

_H015


25/46

Cubic Equation

Comparing co-efficient

2

3

b co efficient of x

a co efficient of x

3

c co efficient of x

a co efficient of x

3

d cons tan t

a co efficient of x

ax3+bx2+cx+d

a[x3-(++)x2+(++)x-()]

_H015


26/46

Cubic Equation

ax3

+bx2

+cx+d=0 a,b,c,dRMaximum real root = ? 3

As degree of equation is 3

Minimum real root? 0?

Complex root occur in conjugate pairwhen co-efficient are real

Maximum no of complex roots=2

Minimum no. of real root is 1

_H015


27/46


If the roots of the equation

x3-2x+4=0 are ,,then thevalue of (1+ ) (1+ ) (1+ ) is

(a)5 (b) 5(c )4 (d) None of these

_H015


28/46


x3-2x+4=0 has roots ,,

a=1, b=0, c=-2, d=4

(1+ ) (1+ ) (1+ )= 1+ + +

b c d1 ( ) ( )

a a a

=1-0-2-4 = -5

If the roots of the equationx3-2x+4=0 are ,,then the value of

(1+ ) (1+ ) (1+ ) is

(a)5 (b)5 (c )4 (d) None of these

Solution Method 1:

_H015


29/46


Method 2

Let f(x)= x3-2x+4

= (x- ) (x- )(x- )for x=-1

f(-1)= 5 = (-1- ) (-1- )(-1- )

(-1)3(1+ ) (1+ )(1+ ) = 5

(1+ ) (1+ )(1+ ) = -5

If the roots of the equationx3-2x+4=0 are ,,then the value of

(1+ ) (1+ ) (1+ ) is

(a) 5 (b) 5 (c )4 (d) None of these

_H015


30/46


31/46

Class Exercise1

If the equations ax2+ bx + c = 0and cx2+ bx + a = 0 haveone root common then

(a)a + b + c = 0(b)a + b c = 0(c) a b + c = 0

(d)both (a) or (c)

Solution:

By observation roots of one equation isreciprocal to other.

So both equation will have common root if it becomes 1or 1


32/46

Class Exercise1

If the equations ax2+ bx + c = 0 andcx2+ bx + a = 0 have one root

common then(a) a + b + c = 0(b) a + b c = 0(c) a b + c = 0(d) both (a) and (c)

When 1 is common root ,a + b + c = 0.

when 1 is common root, a b + c = 0


33/46

Class Exercise2

If x2+ ax + 3 = 0 and

bx2+ 2x + 6 = 0 have both rootscommon then a : b is(a) 2 (b)1/2(c) 4 (d)1/4

Solution:

As both roots are common

1 a 3

b 2 6

1 1

b 2

b 2

a 1a 1

2 2

a 1

b 2


34/46

Class Exercise3

2If x 10x + 22 0, then values of x

are(a) 5 3 x 5 3

(b) x 5 3 or x 5 3

(c) 3 5 x 3 5

(d) None of these

Solution : x210x + 22 0

Factorize into linear terms by using perfectsquare method

22

x 5 3 0


35/46

Class Exercise3


(a) 5 3 x 5 3

(b) x 5 3 or x 5 3

(c) 3 5 x 3 5 (d) None of these

22

x 5 3 0

x 5+ 3 x5 3 0 x - 5 - 3 x - 5+ 3 0

x 5 3 or x 5 3 5-3 5+3

x

Plot on number-line

5-3 and 5-3are included inthe solution set


36/46

Class Exercise4

The number of integral values of

x for which (x 6) (x + 1) < 2(x 9) holds true are

(a) Two (b)Three(c) One (d) Zero

(x 6) (x + 1) < 2 (x 9)

or, x25x 6 < 2x 18 x27x + 12 < 0

or, (x 3) (x 4) < 0 3 < x < 4

So no integral values of x satisfies it.

Solution:


37/46

Class Exercise5

If , , are the roots of

2x3+ 3x22x + 1 = 0. Then thevalue of

is 2 2 2

1 1 1

(a)17/4 (b) 41/4

(c )9/4 (d) None of these

Solution: 2x3+ 3x22x + 1 = 0

3 1

12 2

2 2 2

Now, 1 1 1 2 2 2 2 3


38/46

Class Exercise5

If , , are the roots of 2x3+ 3x22x

+ 1 = 0. Then the value ofis

2 2 2

1 1 1

(a) 17/4 (b) 41/4(c )9/4 (d) None of these

2 2 2

2 3

2

2 2 3

9 32( 1) 2 3

4 2

41

4


39/46

Class Exercise6

If ax2+ bx + c = 0 &

bx2+ c x + a = 0 has acommon root and a 0 thenprove that a3+b3+c3=3abc

Solution:

2

2 2 2

1

ab c bc a ac b

2

2

a b c 0

b c a 0

2 2

22 2ab c bc a;ac b ac b

(bc a2)2= (ab c2) (ac b2)


40/46

Class Exercise6

If ax2+ bx + c = 0 & bx2+ c x + a = 0has a common root and a 0 then prove

that a3+b3+c3=3abc

(bc a2)2= (ab c2) (ac b2)

b2c2+ a42a2bc = a2bc ab3ac3+ b2c2

By expansion:

a(a3+ b3+ c3) = 3a2bc

a(a3+ b3+ c33abc) = 0

either a = 0 or a3+ b3+ c3= 3abc

As a 0

a3+ b3+ c3= 3abc


41/46

Class Exercise7

Find the cubic equation with real

co-efficient whose two roots aregiven as 1 and (1 + i)

Solution:

Imaginary roots occur in conjugate pair; whenco-efficients are real

Roots are 1, (1 i) (1 + i)

Equation is x 1 x 1 i x 1 i 0

or, (x 1) (x22x + 2) = 0

x33x2+ 4x 2 = 0


42/46

Class Exercise8

If ax3+ bx2+ cx + d = 0 has

roots ,and and , then findthe equation whose roots are

2, 2 and 2

Solution:As roots are incremented by 2.

So desired equation can be found replacing x by(x2)

3 2 2a(x 6x 12x 8) b(x 4x 4) c(x 2) d 0

3 2ax (b 6a)x (c 4b 12)x (d 2c 4b 8a) 0

a(x 2)

3

+ b(x 2)

2

+ c(x 2) + d = 0


43/46

Class Exercise9

Find values of x for which theinequation

(2x 1) (x 2) > (x 3) (x 4)holds true

Solution:

(2x 1) (x 2) >(x 3) (x 4)

2x25x + 2 > x27x + 12

x2 + 2x 10 > 0 2 2(x 1) ( 11) 0

x 1 11 x 1 11 0

x 1 11 x 1 11 0


44/46

Class Exercise9

Find values of x for which the inequation(2x 1) (x 2) > (x 3) (x 4) holdstrue

x 1 11 x 1 11 0

Either x 1 11 or x 1 11


45/46

Class Exercise10

For what values of a ,

a(x-1)(x-2)>0 when 1 < x < 2

(a) a > 0 (b) a < 0(c) a = 0 (d) a = 1

Solution:

When 1 < x < 2

(x 1) > 0, (x 2) < 0 (x 1) (x 2) 0

As a (x 1) (x 2) > 0

a < 0


46/46

Thank you

03. quadratic equations-3

Documents