unit 1 quadratic equation examples

7/30/2019 Unit 1 Quadratic Equation Examples

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Mudassar Nazar Notes Page 1

Unit 1 Quadratic Equation

Examples

Example # 1

Solve the quadratic equation 3x2 6x = x + 20 by factorization

Solution

3x2 6x = x + 20

3x2 6x x 20 = 0

3x2

7x

20 = 0

3x2

+ 5x 12x 20 = 0

x ( 3x + 5) 4 ( 3x + 5 ) = 0

( x 4 ) ( 3x + 5 ) = 0

x 4 = 0 or 3x + 5 = 0

x = 0 + 4 or 3x = 0 5

x = 4 or 3x = - 5

x = 4 or x =

S.S =

Example # 2

Solve 5x2

= 30x by factorization

Solution

5x2

= 30x

5x2 30x = 0


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5x( x 6 ) = 0

5x = 0 or x 6 = 0

x = or x = 0 + 6

x = 0 or x = 6

S.S =

Example # 3

Solve the equation x2 3x 4 = 0 by completing square

Solution

x2 3x 4 = 0

x2 3x = 0 + 4

x2 3x + = 4 +

= 4 +

=

=

=

x =

x = or x =

x = or x =


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x = or x =

x = or x =

x = 4 or x = -1

S.S =

Example # 4

Solve the equation 2x2 5x 3 = 0 by completing square.

Solution

2x2 5x 3 = 0

- - = 0

x2

- = o +

x2

- =

x2 + = +

= +

=

=

=

x =


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x = or x =

x = or x =

x = or x =

x = or x =

x = 3 or x =

S.S =

Example # 5

Solve the quadratic equation 2 + 9x = 5x2

by using quadratic formula.

Solution

2 + 9x = 5x2

0 = 5x2 9x 25x2 9x 2 = 0

Comparing it withax2 + bx + c = 0

a = 5 , b = -9 , c = -2By Quadratic formula

x =

x =


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x =

x =

x =

x =

x = or x =x = or x =

x = 2 or x =S.S = {2 , }

Example # 6

Solve the equation - = 0 by using quadratic formula.

Solution

= 0 +

=

(2x + 1) ( x + 4 ) = ( x 2 ) ( x + 2)

2x( x + 4) + 1 ( x + 4 ) = x2

4

2x2

+ 8x + x + 4 = x2 4

2x2

+ 9x + 4 = x2 4

2x2

- x2

+ 9x + 4 + 4 = 0


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x2

+ 9x + 8 = 0

x2 + 9x + 8 = 0Comparing it with

ax2 + bx + c = 0a = 1 , b = 9 , c = 8

By Quadratic formula

x =

x =

x =

x =

x =

x =

x = or x =x = or x =x = -1 or x = -8S.S = {-1 , -8 }


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Example # 7

Solve the equation x4 13x2 + 36 = 0

Solution

x4 13x

2+ 36 = 0

Put x2 = y

Then

x4

= y2

y2 13y + 36 =0

y2

4y

9y + 36 = 0

y2 ( y 4) -9(y 4) = 0

(y 4 ) (y 9) = 0

y 4 = 0 or y 9 = 0

y = 0 + 4 or y = 0 + 9

y = 4 or y = 9

x2 = 4 or x2 = 9

= or =

x = 2 or x = 3

S.S =

Example # 8

Solve the equation 2(2x 1 ) + = 0

Solution


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2(2x 1 ) + = 0 (i)

Let

2x 1 = y

2y + = 5

= 5

2y2

+ 3 = 5y

2y2 5y + 3= 0

2y22y 3y + 3 = 0

2y ( y 1) -3 ( y 1) =0

(y 1) ( 2y 3 ) = 0

y 1 = 0 or 2y 3 = 0

y = 0 + 1 or 2y = 0 + 3

y = 1 or 2y = 3

y = 1 or y =

2x 1 = 1 or 2x 1 =

2x = 1 + 1 or 2x -1 =

2x = 2 or 2x =

x = or 2x =

x = 1 or 2x =

x = 1 or x =


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x = 1 or x =

S.S =

Example # 9

Solve the equation 2x4 5x

3 14x

2 5x + 2 = 0

Solution

2x4 5x

3 14x

2 5x + 2 = 0

Dividing each term by x2

-

- - + =

2x2 5x 14 - + = 0

2x2

+ 5x - - 14 = 0

2 ( x2

+ )5 ( x + ) 14 = 0 (i)

Let

x + = y

then

= y2

x2

+ + 2 = y2

x2

+ = y2 2


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Putting in (i)

2(y2 -2 ) 5y 14 = 0

2y2 4 5y 14 = 0

2y2

5y

18 = 0

2y2

+ 4y 9y 18 = 0

2y ( y + 2) - 9 ( y + 2) = 0

(y + 2) ( 2y 9) = 0

y + 2 = 0 or 2y 9 = 0

y = 0 2 or 2y = 0 + 9

y = -2 or 2y = 9

y = -2 or y =

x + = -2

= -2

x2

+ 1 = -2x

x2 +2x + 1 = 0

x2

+ x + x +1 = 0

x(x +1) + 1 ( x + 1) = 0

( x + 1) ( x + 1) = 0

X + 1 = 0 or x+ 1 = 0

X = 0 1 or x = 01

X = -1 or x = -1

x + =

=


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2(x2

+ 1 ) = 9x

2x2 + 2 = 9x

2x2 9x + 2 = 0

Comparing it withax2 + bx + c = 0a = 2 , b = -9 , c = 2

By Quadratic formula

x =

x =

x =

x =

x = or x =S.S = {-1, , }

Example # 10

Solve the equation 51+x

+ 51-x

= 26

Solution

51+x

+ 51-x

= 26

5 . 5x

+ 5 . 5-x

26 = 0

5 . 5x + - 26 = 0


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Let

5x = y

5y + - 26 = 0

= 0

5y2 26y + 5 = 0 y

5y2 y 25y + 5 = 0

y( 5y 1) 5( 5y 1) = 0

(y 5) ( 5y 1) = 0

y - 5 = 0 or 5y 1 = 0

y = 0 + 5 or 5y = 0 + 1

y = 5 or 5y = 1

y = 5 or y =

5x

= 51

or 5x

= 5-1

x = 1 or x = -1

S.S = { 1 , -1}

Example # 11

Solve the equation ( x -1 ) ( x + 2) (x + 8) ( x + 5) = 19

Solution

( x -1 ) ( x + 2) (x + 8) ( x + 5) = 19

[(x 1) ( x + 8)] [(x + 2) ( x + 5)] = 19

[x(x + 8) -1( x+ 8)] [ x (x + 5) + 2 ( x + 5) ] 19 = 0


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[x2

+ 8x x 8 ] [ x2

+ 5x + 2x + 10] 19 = 0

(x2 + 7x - 8) ( x2 + 7x + 10) 19 = 0

Let

x2 + 7x = y

then

( y 8 ) ( y + 10) 19 = 0

Y( y + 10) 8( y + 10) 19 = 0

y2

+ 10y 8y 80 19 = 0

y2

+ 2y 99 = 0

y2

9y + 11y

99 = 0

y( y 9) + 11( y -9) = 0

( y + 11) ( y 9) = 0

y + 11 = 0 or y 9 = 0

y = 0 11 or y = 0 + 9

y = -11 or y = 9

x2 + 7x = -11 or x2 + 7x = 9

x2

+ 7x + 11 = 0 or x2

+ 7x -9 = 0

x2

+ 7x + 11 = 0

Comparing it withax2 + bx + c = 0

a = 1 , b = 7 , c = 11By Quadratic formula

x =

x =


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x =

x =

x2 + 7x -9 = 0Comparing it with

ax2 + bx + c = 0a = 1 , b = 7 , c = -9By Quadratic formula

x =

x =

x =

x =

S .S = { , }

Example # 12

Solve the equation = 2x + 3

Solution

= 2x + 3Taking square of both sides

= (2x + 3)2


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3x + 7 = (2x)2

+ 2(2x)(3) + (3)2

3x + 7 = 4x2 + 12x + 9

0 = 4x2

+ 12x + 9 3x 7

0 = 4x2 + 9x + 2

4x2

+ 9x + 2 = 0

4x2

+ x + 8x + 2 = 0

x(4x + 1) +2(4x +1) = 0

(x + 2) (4x+1) = 0

X + 2 = 0 or 4x + 1 = 0

X = 0

2 or 4x = 0 -1

X = -2 or 4x = -1

X = -2 or x =

Checking

Put x = -2 in (i)

= 2 (-2) + 3

= -4 + 3

= -1

1 -1 ( False)

So. -2 is an Extraneous Root

Put x =

= 2( ) + 3

= + 3

=


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=

= (True)

S.S =

Example # 13

Solve the equation + =

Solution

+ =

Squaring both sides

=

+ 2 + = x + 11

x + 3 + 2 + x + 6 = x + 11

x + 3 + x + 6 + 2 = x + 11

2x + 9 + 2 - x - 11= 0

x 2 + 2 = 0

2 = 2 x

Taking square of both sides

= (2 x )2

4(x2

+ 9x + 18)= 4 4x + x2

4 x2+ 36x + 72 = 4 4x + x

2


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4 x2x

2+ 36x + 4x + 72 4 = 0

3x2 + 40x + 68 = 0

3x2

+ 6x + 34x + 68 = 0

3x( x + 2) + 34( x + 2) = 0

(x + 2) ( 3x + 34) = 0

x + 2 = 0 or 3x + 34 = 0

x = 0 2 or 3x = 0 34

x = -2 or 3x = -34

x = - 2 or x =

Checking

Put x = -2 in (i)

+ =

+ =

1 + 2 = 3

3 = 3 ( True)

Put x = in ( i)

+ =

+ =

+ =

+ =

( not true)

So, is Extraneous Root

S.S = { -2}


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Example # 14

Solve the equation - = 3

Solution

- = 0

Let

x2 3x = y

Then

=3

= (3)2

+ - 2 = 9

y + 36 + y + 9 - 2 = 9

2y + 45 - 2 = 9

-2 = 9 2y - 45

- 2 = 0 - 2y 36

- 2 = - 2(y 18)

= y + 18

Again squaring both sides

= (y + 18)2

y2 + 45y + 324 = (y)2 + ( 18)2+ 2(y)(18)

y2 + 45y + 324 = y2 + 324 + 36y

y2 + 45y + 324 - y2 - 324 - 36y = 0

45y 36y = 0


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9y = 0

y =

y = 0

So,

x2 3x = 0

x(x 3 ) = 0

x = 0 or x -3= 0

x = 0 or x = 0 + 3

x = 0 or x = 3

S.S = { 0 , 3 }

unit 1 quadratic equation examples

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