section 8.2 the quadratic formula the quadratic formula solving equations using the qf solving an...
TRANSCRIPT
8.2 1
Section 8.2 The Quadratic Formula The Quadratic Formula Solving Equations Using the QF Solving an Equivalent Equation Solving Functions and Rational Equations Problem Solving
a
acbbx
2
42
8.2 2
Introducing … The Quadratic Formula!
The Quadratic Formula is used to find solutions to any quadratic equation
The formula was derived using completing the square and the square root property.
a
acbbxasshownUsually
a
acbbxand
a
acbbx
aresolutionstwothecbxaxFor
2
4
2
4
2
4
0
2
22
2
8.2 3a
acbbx
a
acb
a
bx
a
acb
a
bx
a
acb
a
bx
a
acb
a
bx
a
a
a
c
a
b
a
b
a
bxx
a
c
a
bx
a
ax
cbxax
cbxax
2
4
2
4
2
2
4
2
4
4
2
4
4
2
4
4
42
0
2
2
2
2
22
2
22
2
222
2
2
2
On your next test, Derive the Quadratic Formula !
Move c to the right side
Divide all terms by a
Using b/a, complete the square
Make common denominators
Rewrite in squares format
Take square root of both sides
Remove radicals where possible
Move b/2a to the other side
Combine fractions
8.2 4
Practice with the QF – Solve:
362
93
2
813
2
7293
)1(2
)18)(1(4)3(3
1831
0183
293
293
2
2
xorx
x
x
cba
xx
2
22
4
224
4
8164
)2(2
)1)(2(4)4(4
142
0142
142
2
2
2
x
x
x
cba
xx
xx
8.2 5
Solving functions
535
2
53210
2
21210
)1(2
)28)(1(4)10(10
28101
02810
7)4(74
4)4
771(
4
77)(
1)(
2
2
2
x
x
x
cba
xx
xxxx
xxxx
xxxf
xfthatsuchxallFind
8.2 6
ImaginarySolutions
ixandix
ix
x
cba
xx
xxx
xxx
27
21
27
21
2
2
2
2
71
2
71
)1(2
)2)(1(411
211
02
245
)12(2)5(
8.2 7
Solving Equivalent Equations Sometimes we need to simplify first
0,2
22
4
224
4
84
4
8164
)2(2
)1)(2(444
1,4,2)0(
0)142(
042
2
2
23
xx
x
x
x
cbax
xxx
xxx
15
1116
30
44412
)15(2
)5)(15(41212
5,12,15
051215
15)0(
2
2
31
542
x
x
x
cba
xx
xx
2
173
)1(2
)2)(1(4)3()3(
2,3,1
023
)0406020(
2
2
2012
x
x
cba
xx
xx
8.2 8
What Next? Discriminants! Present Section 8.3 Studying QE
Solutions