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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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Mudassar Nazar Notes Page 2
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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Mudassar Nazar Notes Page 3
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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Mudassar Nazar Notes Page 4
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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Mudassar Nazar Notes Page 5
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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Mudassar Nazar Notes Page 6
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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Mudassar Nazar Notes Page 7
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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Mudassar Nazar Notes Page 8
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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Mudassar Nazar Notes Page 9
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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Mudassar Nazar Notes Page 10
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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Mudassar Nazar Notes Page 11
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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Mudassar Nazar Notes Page 12
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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Mudassar Nazar Notes Page 13
[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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Mudassar Nazar Notes Page 14
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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Mudassar Nazar Notes Page 15
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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Mudassar Nazar Notes Page 16
=
= (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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Mudassar Nazar Notes Page 17
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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Mudassar Nazar Notes Page 18
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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Mudassar Nazar Notes Page 19
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 }