Post on 22-Jan-2016
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M3U3D5 Warm UpM3U3D5 Warm UpMultiply:
(-5 + 2i)(6 – 4i)
-30 +20i + 12i – 8i2
-30 + 32i – 8(-1)-30 + 32i + 8
-22 + 32i
F O I L
Remember the cycle of “i”
i0 = 1 i2 = -1i1 = i i3 = -i
Homework Check:
Homework Check:
Quadratic Formula, Discriminant, and Zeroes
OBJ: Solve quadratic equations with real coefficients that have
complex solutions.
THE QUADRATIC FORMULATHE QUADRATIC FORMULA
1. When you solve using completing the square on the general formula you get:
2. This is the quadratic formula!
3. Just identify a, b, and c then substitute into the
formula.
2 4
2
b b acx
a
2 0ax bx c
2 4
2x
cb b a
a
The Quadratic Formula Derived
02 cbxax
cbxax 2
a
cx
a
bx
a
a 2
a
cx
a
bx
2
a
b
a
b
22
1 2
22
42 a
b
a
b
a
c
a
b
a
bx
a
bx
2
2
2
22
44
a
a
a
c
a
b
a
bx
a
bx
4
4
44 2
2
2
22
2 4
2x
cb b a
a
The Quadratic Formula Derived
22
2
2
22
4
4
44 a
ac
a
b
a
bx
a
bx
2
2
2
22
4
4
4 a
acb
a
bx
a
bx
2
2
2
22
4
4
4 a
acb
a
bx
a
bx
2
22
4
4
2 a
acb
a
bx
2
2
4
4
2 a
acb
a
bx
a
acb
a
bx
2
4
2
2
a
acb
a
bx
2
4
2
2
a
acbbx
2
42
The quadratic formula is used to solve any quadratic equation.
2 4
2x
cb b a
a
The quadratic formula is:
What are a, b, and c?
Standard form of a quadratic equation is: 2 0x xba c
2 4 8 0x x
a 1 c b4 8
23 5 6 0x x
a 3 c b 5
22 0x x
a 2 c b1 0
2 10x a 1 c b0 106
2 10 0x
The Quadratic Formula
2 4
2x
cb b a
a
2 0x xba c
2 3 2 0x x
2x 1x
1x 2x 0
1 0x 2 0x
The Quadratic Formula
Solve the problem by factoring.
2 4
2x
cb b a
a
2 0x xba c
2 3 2 0x x a 1 c b 3 2
23 3 1 24
12x
3 9 8
2x
3 1
2x
3 1
2x
3 1
2x
3 1
2x
4
2x
2x
2
2x
1x 3 1
2x
Solve the same problem using The Quadratic Formula
2 4
2x
cb b a
a
2 0x xba c
22 5 0x x
a 2 c b 1 5
24
22
1 521x
1 1 40
4x
1 41
4x
1. You try.
2 4
2x
cb b a
a
2. Another example.
44 2 xx
044 2 xx
42
44411 2 x
8
6411 x
8
631 x
8
631 ix
8
791
ix
8
731 ix
ix
8
73
8
1
WHY USE THE WHY USE THE
QUADRATIC FORMULA?QUADRATIC FORMULA?
• The quadratic formula allows you to solve ANY
quadratic equation, even if you cannot factor it.
• An important piece of the quadratic formula is
what’s under the radical:
b2 – 4ac
• This piece is called the discriminant.
WHY IS THE DISCRIMINANT WHY IS THE DISCRIMINANT
IMPORTANT?IMPORTANT?
The discriminant tells you the number and types of answers
(roots) you will get. The discriminant can be +, –, or 0
which actually tells you a lot! Since the discriminant is
under a radical, think about what it means if you have a
positive or negative number or 0 under the radical.
WHAT THE DISCRIMINANT WHAT THE DISCRIMINANT
TELLS YOU!TELLS YOU!
Value of the Discriminant
Nature of the Solutions
Negative 2 imaginary solutions
Zero 1 Real Solution
Positive – perfect square 2 Reals- Rational
Positive – non-perfect square
2 Reals- Irrational
Example #1Example #1
22 7 11 0x x
Find the value of the discriminant and describe the nature of the roots (real, imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula)
1. a=2, b=7, c=-11
Discriminant = 2
2
4
(7) 4(2)( 11)
49
137
88
b ac
Discriminant =
Value of discriminant=137
Positive-NON perfect square
Nature of the Roots – 2 Reals - Irrational
Example #1- continuedExample #1- continued
22 7 11 0x x
2
2
4
2
7 7 4(2)( 11)
2(
2, 7, 11
7 137 2 Reals - Irrational
4
2)
a b
b ac
a
c
b
Solve using the Quadratic Formula
Solving Quadratic Equations Solving Quadratic Equations
by the Quadratic Formulaby the Quadratic Formula
2
2
2
2
2
1. 2 63 0
2. 8 84 0
3. 5 24 0
4. 7 13 0
5. 3 5 6 0
x x
x x
x x
x x
x x
Try the following examples. Do your work on your paper and then check your answers.
1. 9,7
2.(6, 14)
3. 3,8
7 34.
2
5 475.
6
i
i
2 4
2x
cb b a
a
The Quadratic Formula and the Discriminant
REMEMBER, the discriminant is the radicand portion of the quadratic formula (b2 – 4ac).
It is used to discriminate among the possible number and type of solutions a quadratic equation will have.
b2 – 4acName and Type of
SolutionPositive
Zero
Negative
Two real solutions
One real solutions
Two complex, non-real solutions
2 4
2x
cb b a
a
The Quadratic Formula and the Discriminate
2143 2
89
b2 – 4acName and Type of
SolutionPositive
Zero
Negative
Two real solutions
One real solutions
Two complex, non-real solutions
2 3 2 0x x a 1 c b 3 2
1
Positive
Two real solutions
2x 1x
2 4
2x
cb b a
a
The Quadratic Formula and the Discriminate
4441 2
641
b2 – 4acName and Type of
SolutionPositive
Zero
Negative
Two real solutions
One real solutions
Two complex, non-real solutions
a c b
63
Negative
Two complex, non-real solutions
044 2 xx
4 1 4
ix8
73
8
1
2(25)
4(25)(36)- 60)(- 60)(- - x
2
25x 25x 22 - 60x + 36 = 0 - 60x + 36 = 0
18 x 30- x 225 2 f(x)
Find the zeros using the Find the zeros using the
Quadratic FormulaQuadratic Formula
2(25)
4(25)(36)- 60)(- 60)(- - x
2
Exact Solution:Exact Solution:
56
50
0 60 x
2(25)
4(25)(36)- 60)(- 60)(- - x
2
Calculator Calculator Solution:Solution: x = 1.2x = 1.2
Check Check Intercepts!Intercepts!
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
f(x) = 3x f(x) = 3x 22 + 2 - 4x + 2 - 4x
3x 3x 22 - 4x + 2 = 0 - 4x + 2 = 0
2(3)
4(3)(2)- 4)(- 4)(- - x
2
Find the zeros using the Find the zeros using the
Quadratic Formula…You try!Quadratic Formula…You try!
2(3)
4(3)(2)- 4)(- 4)(- - x
2
6
8- 4 x
No real solution. No real solution.
Check Intercepts.Check Intercepts.
2 4
2x
cb b a
a
A real world application of The Quadratic Formula:
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
20 feet
x + 2
x
2 4
2x
cb b a
a
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2 + b2 = c2
(x + 2)2 + x2 = 202
x2 + 4x + 4 + x2 = 400
2x2 + 4x + 4 = 400
2x2 + 4x – 396 = 0
2(x2 + 2x – 198) = 0
2 4
2x
cb b a
a
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2 + b2 = c2
2(x2 + 2x – 198) = 0
12
1981422 2 x
2
79242 x
2
7962 x
2 4
2x
cb b a
a
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2 + b2 = c2
2
7962x
2
2.282
2
2.282 x
2
2.282 x
2
2.26x
1.13x
2
2.30x
1.15xft
2 4
2x
cb b a
a
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2 + b2 = c2
1.13x
ft2.28
ft
21.131.13
28 – 20 = 8 ft
ClassworkClasswork U3D5 Packet Pages 1 & 2
HomeworkHomeworkU3D5 Packet Pages 3 & 4 odds