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Title: Algebra _ Quadratic Equations
Description: Algebra _ Quadratic Equations To solve a quadratic equation by factoring, Put all terms on one side of the equal sign, leaving zero on the other side. Factor. Set each factor equal to zero. Solve each of these equations. Check by inserting your answer in the original equation.
Description: Algebra _ Quadratic Equations To solve a quadratic equation by factoring, Put all terms on one side of the equal sign, leaving zero on the other side. Factor. Set each factor equal to zero. Solve each of these equations. Check by inserting your answer in the original equation.
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Algebra
Quadratic Equations
completion of square method
Solution of quadratic equation by formula method
I
...
x2 + 10x + 9 = 0
Solution:
x2 + 10x + 25 – 25 + 9 = 0
( x + 5)2 – 16 = 0
( x + 5)2 = 16
I
...
5x2 + 14x = 55
Solution:
5x2 + 14x = 55
5
x+
14
x2 + 5 x 11 = 0
7
5
2
49
11 = 0
25
x+ 7
5
x+
7
5
2
=
2
=
7
x+ 5
x
18
5
+ 11 =
324
25
2
=
18
5
=
18
7
5
5
=
11
5
The roots are 11 and – 5
5
49
25
(or)
x +
7
5
=
x
=
x =
18
5
18
7
5
5
25
5 = 5
I
...
x2 – 4x – 45 = 0
Solution:
(x – 2)2 – 4 – 45 =
0
( x 2)2 = 49
x–2 =
x–2
x
I
...
15 = 17x + 4x2
Solution:
4
4x2 + 17x – 15 = 0
15
x2 + 17 x –
= 0
4
4
17 2 289
15
x + 8 64 4
x+
17
8
17
x+
8
x+
= 0
2
=
289
15
+
64
4
2
17
8
x
x
=
23
8
= +
23
8
=
529
64
2
=
23
17
8
8
=
6
3
=
8
4
=
3
,5
4
(or) x +
17
23
=
8
8
(or) x =
23
8
40
8
17
8
= 5
I
...
5x + 7
= 3x + 2
x–1
Solution:
5x + 7
3
3x2 – 6x – 9
= 0
x2 – 2x – 3
= 0
(x – 1)2 – 1 – 3
= 0
2
( x – 1)
= 4
( x – 1)
= 2
x–1
x
II
...
x2 + 2x – 2 = 0
Solution:
a = 1, b = 2, c = 2
x
=
b –b2 – 4ac
2a
x
=
2 – 4 + 8
2
x
= 1 –3
=
2 –12
2
=
2 2–3
2
II
...
x2 – 6x – 3 = 0
Solution:
II
...
2x2 3x – 5 = 0
Solution:
a = 2,
x
b = 3, c = 5
=
=
x =
II
...
4x2 + 7x + 2 = 0
Solution:
a = 4,
b = , c = 2
–49 – 32
x =
8
=
7 17
8
6 4 –3
2
II
...
(x – 3)2 = 2 (x + 4)
Solution:
x2 – 6x + 9 – 2x – 8
= 0
x2 – 8x + 1
= 0
a = 1, b = 8, c = 1
x
II
...
3x2 + 2 5 x – 5 = 0
a = 3, b = 2 5 , c = 5
Solution:
x
=
2 5 20 + 60 2
5 80
=
6
6
=
2 5 45
6
=
2 5
6
or
65
6
x = 5 or
5
3
II
...
– x + 5 =
2x + 3
Solution:
Squaring
= (2x + 3)2
x+5
x + 5 = 4x2 + 12x + 9
4x2 + 11x + 4 = 0
a = 4, b = 11, c = 4
x
=
11 –121 – 64
8
=
11 – 57
8
II
...
a (x2 + 1) = x (a2 + 1)
Solution:
ax2 – x (a2 + 1) + a = 0
x =
II
...
3 a2 x2 – a b x – 2 b2 = 0
Solution:
x =
=
=
=
ab – a2 b2 + 24 a2 b2
6a2
ab – 25a2 b2
6a2
6ab or
6a2
b
a
or
4ab
6a2
2b
3a
=
ab 5ab
6a2
II
...
4x2 – 2 (a2 + b2) x + a2 b2 = 0
Solution:
x
2(a2 + b2) 4(a2 + b2)2 –16a2b2
=
8
=
2(a2 + b2) 4a4 + 4b4 – 8a2b2
8
=
2(a2 + b2) 2 (a2 – b2)2
8
=
2 (a2 + b2) 2(a2 – b2)
8
=
(a2+ b2) (a2 – b2)
4
2
= 2a or
4
=
II
...
4 x2 – 4 a2 x + (a4 – b4) = 0
Solution:
x =
4a2 16a2 – 16(a4– b4)
8
=
4a2 16b4 =
8
=
a2 b2
2
4a2 4b2
8
a2 + b2
2
2
2
, a –b
2
II
...
p2 x2 + (p2 – q2) x – q2 = 0
Solution:
(p2 – q2) ( p2– q2)2 + 4 p2 q2
=
x
2p2
(p2– q2) p4 + a4 2 p 2q2 + 4 p2 q2
=
=
2p2
(p2– q2) p4 + q4 + 2p2 q2
=
2p2
(p2 + q2) (p2– q2)
2p2
2p2
= 2 or
2p
= 1 or
II
...
36 x2 – 12 a x + (a2 – b2) = 0
Solution:
x =
12a 144a2 – 144(a2 – b2)
72
12a 144b2
12a 12b
=
72
72
a + b or
a–b
= a b
6
6
6
=
Title: Algebra _ Quadratic Equations
Description: Algebra _ Quadratic Equations To solve a quadratic equation by factoring, Put all terms on one side of the equal sign, leaving zero on the other side. Factor. Set each factor equal to zero. Solve each of these equations. Check by inserting your answer in the original equation.
Description: Algebra _ Quadratic Equations To solve a quadratic equation by factoring, Put all terms on one side of the equal sign, leaving zero on the other side. Factor. Set each factor equal to zero. Solve each of these equations. Check by inserting your answer in the original equation.