Download 4-6 Perform Operations with Complex Numbers

4-6 Perform Operations with Complex Numbers
Name:__________________
Objective: To use complex numbers to perform operations.
Algebra 2 Standards 5.0, 6.0, and 8.0
* Sometimes a quadratic equation does not have a real-number solution(Ex: x2 = -1).
To overcome this, there is an expanded system of numbers using the imaginary unit (i),
defined as ________________.
The imaginary unit i can be used to write the square root of any negative number.
*The Square Root of a Negative Number
Property
1. If r is a positive real number, then
Example
__________________________________
2. By Property (1), it follows that
__________________________________
Ex. 1: Solve
a. x2 = -13
b.
You Try: Solve
a. x2 = -38
b. 3x2 – 7 = -31
2x2 + 18 = -72
*A Complex Number written in standard form is a number _______________ where a and b
are real numbers. The number a is the real part of the complex number and the number bi is the
imaginary part.
Complex Numbers
Real Numbers
(a + 0i)
-1

5
2
2
Imaginary Numbers
(a + bi, b  0 )
Pure Imaginary
Numbers
(0 + bi, b  0 )
-4i
Alg 2 Ch 4B Notes Page 1
5 – 5i
2 + 3i
6i
If b  0 then a + bi is an imaginary
number. If a = 0 and b  0 then a +
bi is a pure imaginary number.
Two complex numbers a + bi and c
+ di are equal if and only if a = c
and b = d.
*Sums and Differences of Complex Numbers: To add or subtract two complex numbers, add
or subtract their real parts and their imaginary parts separately.
Sum of complex numbers:
(a + bi) + (c + di) = (a + c) + (b + d)i
Difference of complex numbers: (a + bi) – (c + di) = (a – c) + (b – d)i
Ex. 2: Write the expression as a complex number in standard form.
a. (12 – 11i) + (-8 + 3i)
b. (15 – 9i) – (24 – 9i)
c.
35 – (13 + 4i) + i
You Try: Write the expression as a complex number in standard form.
a. (9 – i) + (-6 + 7i)
b. (3 + 7i) – (8 – 2i)
c. -4 – (1 + i) – (5 + 9i)
*Multiplying Complex Numbers:
To multiply two complex numbers, use the Distributive Property or the FOIL method.
Ex. 3: Write the expression as a complex number in standard form.
a.
-5i(8 – 9i)
b.
(-8 + 2i)(4 – 7i)
c.
6 10
You Try: Write the expression as a complex number in standard form.
4 9
a.
i(7-3i)
b.
(6 + i)(5- 3i)
c.
Alg 2 Ch 4B Notes Page 2
*Complex Conjugates:
Two complex numbers of the form a + bi and a – bi are called complex conjugates.
The product of complex conjugates is always a _____________ number.
We can use conjugates to write the quotient of two complex numbers in standard form.
Ex. 4. Write the expression in standard form.
5
10  2i
a.
b.
7i
1 i
You Try: Write the expression in standard form.
6
10  3i
a.
b.
3  2i
5i
.
c.
3  4i
5i
c.
5  3i
4i
*Complex Plane: Every complex number corresponds to a point in the complex plane.
The complex plane has a horizontal axis called the ________________ axis and a vertical axis
called the ______________________ axis
Ex. 5: Plot the complex numbers in the same complex plane.
a.
4 + 2i (start at the origin, move right 4 and up 2)
b.
-1 + 3i
c.
-4i
d.
2 – 2i
Alg 2 Ch 4B Notes Page 3
You Try: Plot the complex numbers in the same complex plane.
a.
4–i
b.
-3 – 4i
c.
2 – 5i
d.
3i
*Absolute Value of a Complex Number:
The absolute value of a complex number
z = a + bi, denoted z ,
is a nonnegative real number defined as z  a 2  b2 .
This is the _____________________
between z and the origin in the complex plane.
Ex. 6: Find the absolute value of each.
a. 5 – 12i
b.
You Try: Find the absolute value of each.
a. 4 – i
b. – 4i
c. 2 + 5i
Alg 2 Ch 4B Notes Page 4
17i
d. –2 + 3i
4-7 Complete the Square
Name:_______________
Objective: To solve quadratic equations by completing the square.
Algebra 2 Standards 8.0 and 10.0
*Recall: We can solve x 2  k by taking the square root.
If one side of an equation is a perfect square trinomial, we can also solve it by taking the square
root.
Ex. 1: Solve x2 + 20x + 100 = 81
You Try: Solve x2 – 10x + 25 = 1
*Completing the Square:
In example 1, x2 + 20x + 100 is a perfect square trinomial because it equals (x + 10)2.
Sometimes you need to add a term to an expression _________ to make it a square.
This process is called Completing the Square.
Words: To complete the square for the expression x2 + bx, add ________.
Algebra:
b
Steps: 1) Find half of x-term coefficient and square it : ( ) 2
2
b
2) Add ( ) 2 and write as a binomial squared.
2
Ex. 2: Find the value of c that makes x2 – 28x + c a perfect square trinomial. Then write the
expression as the square of a trinomial.
You Try: Find the value of c that makes x2 + 14x + c a perfect square trinomial. Then write the
expression as the square of a trinomial.
*Solving equations: You can use completing the square to solve any quadratic equation. You
must be sure to add the same number to both sides of the equation.
Alg 2 Ch 4B Notes Page 5
*Solve ax2 + bx+ c = 0 when a = 1 using completing the square
Ex.) x2 – 10x + 1 = 0
1) Move constant to the right and leave space
1
2) Take b , square it, and add that value to both sides
2
b
3) Factor left side as ( x  ) 2 and simplify the right side.
2
4) Take square root both sides (  !)
5) Solve for x
Example 3: Solve x 2  8 x  11  0 by completing the square.
You try: Solve x2 + 6x + 4 = 0
by completing the square.
You Try: Solve x2 – 12x + 8 = 0
*Solve ax2 + bx+ c = 0 when a ≠ 1
To complete the square, you must divide everything by a so that a  1
Ex. 4: Solve 3x2 – 36x + 150 = 0 by completing the square.
Alg 2 Ch 4B Notes Page 6
You Try: Solve 2x2 – 4x – 14 = 0
Ex. 5: Solve 6x(x + 8) = 12 by completing the square.
*Vertex Form:
Recall that the vertex form of a quadratic function is ______________________ where (h, k)
is the ______________ of the function’s graph. We can use completing the square to write a
quadratic function in vertex form.
Ex. 6: Write y = x2 + 18x + 95 in vertex form. Then identify the vertex.
(steps)
 Write the original function
 Move constant to y side
b
 Add ( ) 2 to each side
2
 Write trinomial as a binomial squared
 Solve for y
You Try: Write y = x2 – 8x + 17 in vertex form. Then identify the vertex.
Ex. 7: The height y (in feet) of a ball that was thrown up in the air from the roof of a building
after t seconds is given by the equation y = -16t2 + 64t + 50. Find the maximum height of the
ball.
You Try: Write y = 2x2 – 4x – 4 in vertex form. Then identify the vertex.
Alg 2 Ch 4B Notes Page 7
**Deriving the Quadratic Formula by Completing the Square
We will develop the Quadratic Formula by starting with the standard form of a quadratic
equation and then solve for x by completing the square.
ax 2  bx  c  0 ,
Step 1
where a  0 .
ax 2  bx  c  0 , a  0
Step 2
x2 
b
c
x
a
a
Step 3
x2 
b
c  b 
 b 
x     
a
a  2a 
 2a 
2
2
Step 4
b 
c b2

x 
   2
2a 
a 4a

Step 5
b 
c 4a b 2


x 
  
2a 
a 4a 4a 2

Step 6
b 
b 2  4ac

x




2a 
4a 2

2
2
b 
b 2  4ac

x
 
2a 
4a 2

2
Step 7
Step 8
b 
b 2  4ac

x 

2a 
4a 2

Step 9
x
Step 10
x
Step 11
x
Alg 2 Ch 4B Notes Page 8
b
b 2  4ac

2a
2a
b
b 2  4ac

2a
2a
 b  b 2  4ac
2a
2
Ex: 4 x 2  8 x  7
4-8 Use the Quadratic Formula and the Discriminant
Name:______________
Objective: To solve quadratic equations using the quadratic formula.
Algebra 2 Standard 8.0
*The Quadratic Formula: Let a, b, and c be real numbers such that a ≠ 0.
The solutions of the quadratic equation ax2 + bx + c = 0 are
_________________________________.
---Be sure that the original equation is in standard form.
Ex. 1: Solve x2 – 5x = 7
You Try: Solve x2 = 6x – 4
Ex. 2: Solve 16x2 – 23x = 17x – 25
You Try: Solve 4x2 – 10x = 2x – 9
*In Algebra 1, if we got a negative number inside the radical, we said the equation had no real
solutions. Now we will take the equation further using imaginary solutions.
Ex. 3: Solve x2 – 6x + 10 = 0
Alg 2 Ch 4B Notes Page 9
You Try: 7x – 5x2 – 4 = 2x + 3
*Using the Discriminant of ax2 + bx + c = 0
The Quadratic Formula: x 
Value of the
discriminant
Number and type
of solutions
b  b2  4ac
,
2a
b2 – 4ac is called the Discriminant.
b2 – 4ac > 0
b2 – 4ac = 0
b2 – 4ac < 0
______ real solutions
_____ real solution
Two ______________
solutions
______ x-intercepts
______ x-intercept
______ x-intercepts
Graph of
ax2 + bx + c = 0
Ex. 4: Find the discriminant of the quadratic equation and give the number and type of solutions
of the equation.
a. x2 + 10x + 23 = 0
b. x2 + 10x + 25 = 0
c. x2 + 10x + 27 = 0
You Try: Find the discriminant of the quadratic equation and give the number and type of
solutions of the equation.
a. 3x2 + 12x + 12 = 0
b. 2x2 + 4x – 4 = 0
c. 7x2 – 2x = 5
*Modeling Launched Objects: Recall that the formula for a dropped object is h = -16t2 + h0.
For an object that is launched or thrown, an extra term v0t must be added to the model to account
for the object’s initial vertical (upward) velocity v0 (in feet per second). Recall that h is the
height (in feet), t is the time in motion (in seconds), and h0 is the initial height (in feet).
h = -16t2 + h0
Object is dropped
2
h = -16t + v0t + h0
Object is launched
The value of v0 can be ________________, __________________, or ____________, depending
on whether the object is launched upward, downward, or parallel to the ground.
Ex. 5: A basketball player passes the ball to a teammate. The ball leaves the player’s hand 5 feet
above the ground and has an initial vertical velocity of 55 feet per second. The teammate catches
the ball when it returns to a height of 5 feet. How long was the ball in the air?
Alg 2 Ch 4B Notes Page 10