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John 6:47
Truly, truly, I say to you, whoever believes has
eternal life.
Isaiah 33:22
For the LORD is our judge, the LORD is our
lawgiver, the LORD is our king; he will save
us.
The Quadratic Formula
Quadratic Formula
a
acbbx
2
42
02 cbxax
IF
THEN
Proving the Quadratic Formula
Solve by completing the square
02 cbxax
a
cx
a
bx 2
2
22
42 a
b
a
b
a
c
a
b
a
bx
a
bx
2
2
2
22
44
a
c
a
b
a
bx
2
22
42
2
22
4
4
2 a
acb
a
bx
a
acb
a
bx
2
4
2
2
22
22
4
4
42 a
ac
a
b
a
bx
24LCM a
a
acbbx
2
42
a
acb
a
bx
2
4
2
2
#1 Solve using the quadratic formula.
0273 2 xx
a
acbbx
2
42
2 ,7 ,3 cba
)3(2
)2)(3(4)7()7( 2 x
6
24497 x
6
57 x
6
12x
6
2x
3
1 ,2x
6
257 x
#2 Solve using the quadratic formula542 2 xx
a
acbbx
2
42
5 ,4 ,2 cba
)2(2
)5)(2(4)4()4( 2 x
4
40164 x
4
564x
4
1424 x
0542 2 xx
2
141x
9.2x 9.0x
4
1444 x
Solve using the quadratic formula
3.
4.
5.
752 xx
mm 1083 2
ix2
3
2
5
3
2 ,4 m
xx 16642 8x
#3 Solve using the quadratic formula
752 xx
a
acbbx
2
42
7 ,5 ,1 cba
)1(2
)7)(1(455 2 x
2
28255 x
2
35 x
ix2
3
2
5
0752 xx
ix2
3
2
5
#4 Solve using the quadratic formula
mm 1083 2
a
acbbm
2
42
8 ,10 ,3 cba
)3(2
)8)(3(4)10()10( 2 m
6
9610010 m
6
1410m
6
24m
6
4m
3
2 ,4 m
08103 2 mm
6
19610 m
#5 Solve using the quadratic formula
xx 16642
a
acbbx
2
42
64 ,16 ,1 cba
)1(2
)64)(1(416162
x
2
25625616 x
2
016 x
8x
064162 xx
The Discriminant
Discriminant In the quadratic
formula, the expression
underneath the radical
that describes the
nature of the roots.
Understanding the discriminant
Discriminantacb 42
# of real roots
042 acb 2 real roots
042 acb 1 real roots
042 acb No real roots
#6 Using the discriminant
0134 2 yy
acb 4 nt discrimina 2
1 ,3 ,4 cba
)1)(4(4)3( nt discrimina 2
169 nt discrimina
52 nt discrimina
052
roots real 2
#7 Using the discriminant
54 2 xx
acb 4 nt discrimina 2
5 ,1 ,4 cba
)5)(4(4)1( nt discrimina 2
801 nt discrimina
79 nt discrimina
079
no real roots
Using the discriminant
8.
9.
542 2 xx
484 2 xx
56nt discrimina
roots real 2
0 nt discrimina
root real 1
#8 Using the discriminant
542 2 xx
acb 4 nt discrimina 2
0542 2 xx
)5)(2(4)4( nt discrimina 2
4061 nt discrimina
56nt discrimina
056
roots real 2
#9 Using the discriminant
484 2 xx
acb 4 nt discrimina 2
0484 2 xx
)4)(4(4)8( nt discrimina 2
6446 nt discrimina
0 nt discrimina
00
root real 1
