Materi Mat
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Solving Quadraticsby
Completing the Square&
Quadratic Formula
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: October 25, 2005
X2 + 6x + 9
Jeff Bivin -- LZHS
2)3( x
X2 + 4x + 4
Jeff Bivin -- LZHS
4
2)2( x
X2 + 5x + 25/4
Jeff Bivin -- LZHS
225 )( x
X2 - 6x + 9
Jeff Bivin -- LZHS
2)3( x
Solve by Completing the Square
x2 + 10x + 8 = 0
(x2 + 10x ) = -8
(x2 + 10x + (5)2) = -8 + 25
(x + 5)2 = 17
(5)2 = 25
175 x175 x
51021
Jeff Bivin -- LZHS
Solve by Completing the Square
3x2 + 24x + 12 = 0
(x2 + 8x ) = -4
(x2 + 8x + (4)2) = -4 + 16
(x + 4)2 = 12
(4)2 = 16
124 x124 x
4821
x2 + 8x + 4 = 03
324 xJeff Bivin -- LZHS
Solve by Completing the Square
2x2 + 5x - 12 = 0
411
45 x
45
25
21
2
23
46 x
06252 xx
6)( 252 xx
16252
45
252 6))(( xx 16
252
45
1625
16962
45 )( x 16
121
411
45 )( x
4416 x
Jeff Bivin -- LZHS
Solve by Completing the Square
2x2 - 12x - 11 = 0
2583 x
3621
206 2
112 xx
2112 )6( xx
9))3(6( 21122 xx 93 2
218
2112)3( x 2
29
229)3( x
Jeff Bivin -- LZHS
Solve by Completing the Square
-5x2 + 12x + 19 = 0
5131
56 x
56
512
21
-505
195122 xx
519
5122 )( xx
2536
5192
56
5122 ))(( xx 5
362
56
2536
25952
56 )( x 25
131
5131
56 )( x
51316 x
Jeff Bivin -- LZHS
Solve by Completing the Square
ax2 + bx + c = 0
a
acb
abx 2
4
2
2
ab
ab
221
a02 a
cab xx
ac
ab xx )( 2
2
2
2
222
2 ))((ab
ac
ab
ab xx 2
2
4
2
2 ab
ab
2
2
4
22 )(
ab
ac
abx
2
2
4
4
a
acb
a
acb
abx 4
4
2
2
)(
a
acbbx 2
42 Jeff Bivin -- LZHS
Solve using the Quadratic Formula
3x2 + 7x - 4 = 0
a = 3b = 7c = -4
a
acbbx 2
42
32
)4(3477 2
x
6
48497 x
6977 x
Jeff Bivin -- LZHS
Solve using the Quadratic Formula
6x2 + 9x + 2 = 0
a = 6b = 9c = 2
a
acbbx 2
42
62
26499 2
x
12
48819 x
12339 x
Jeff Bivin -- LZHS
Solve using the Quadratic Formula5x2 - 8x + 1 = 0
a = 5b = -8c = 1
a
acbbx 2
42
52
154)8(8 2
x
10
20648 x
10448 x
101142 5
114 101128 x
Jeff Bivin -- LZHS
Solve using the Quadratic Formula6x2 - 17x - 14 = 0
a = 6b = -17c = -14
a
acbbx 2
42
62
1464)17(17 2
x
12
33628917 x
1262517 x
32
128 x
27
1242 x
122517 x
Jeff Bivin -- LZHS
Solve using the Quadratic Formulax2 + 6x + 9 = 0
a = 1b = 6c = 9
a
acbbx 2
42
12
91466 2
x
2
36366 x
206 x
26x 3
Jeff Bivin -- LZHS
Solve using the Quadratic Formula
3x2 + 7x + 5 = 0
a = 3b = 7c = 5
a
acbbx 2
42
32
53477 2
x
6
60497 x
6117 x
Jeff Bivin -- LZHS
Why do some quadratic equations have 2 real solutions
some have 1 real solution and some have no real
solutions?
Now Consider
ax2 + bx + c = 0
Jeff Bivin -- LZHS
a
acbbx 2
42
If discriminant > 0, then 2 real solutions
If discriminant = 0, then 1 real solution
If discriminant < 0, then 0 real solutions
Jeff Bivin -- LZHS