Download Algebra 2: Unit 5 Continued

Algebra 2: Unit 5 Continued
FACTORING
QUADRATIC
EXPRESSION
Factors
Factors are numbers or expressions that you
multiply to get another number or expression.
Ex. 3 and 4 are factors of 12 because 3x4 = 12
Factors
What are the following expressions factors of?
1. 4 and 5?
2. 5 and (x + 10)
3. 4 and (2x + 3)
4. (x + 3) and (x - 4)
GCF
One way to factor an expression is to factor out a
GCF or a GREATEST COMMON FACTOR.
EX: 4x2 + 20x – 12
EX: 9n2 – 24n
Try Some!
Factor:
a. 9x2 +3x – 18
b. 7p2 + 21
c. 4w2 + 2w
Factors of Quadratic Expressions
When you multiply 2 binomials:
(x + a)(x + b) = x2 + (a +b)x + (ab)
This only works when the coefficient for
x2 is 1.
Finding Factors of Quadratic Expressions
When a = 1: x2 + bx + c
Step 1. Determine the signs of the factors
Step 2. Find 2 numbers that’s product is c,
and who’s sum is b.
Sign table!
2nd
sign
+
Same
Different
Question
1st
sign
Answer
+
-
+ or -
(x+
)(x+
)
(x )(x )
(x + )(x - )
OR
(x - )(x + )
Examples
Factor:
1. x2 + 5x + 6
2. x2 – 10x + 25
3. x2 – 6x – 16
4. x2 + 4x – 45
Examples
Factor:
1. x2 + 6x + 9
2. x2 – 13x + 42
3. x2 – 5x – 66
4. x2 – 16
Box and Slide Method
When a does NOT equal 1.
Steps
1. Slide
2. Factor
3. Divide
4. Reduce
5. Slide
Example!
Factor:
1.
3x2 – 16x + 5
Example!
Factor:
2.
2x2 + 11x + 12
Example!
Factor:
3.
2x2 + 7x – 9
You Try!
Factor
1. 5t2 + 28t + 32
2. 2m2 – 11m + 15
Quadratic Equations
5 ways to solve
There are 5 ways to solve quadratic equations:
1. Factoring
2. Finding the Square Root
3. Graphing
4. Completing the Square
5. Quadratic Formula
Quadratic Equation
Standard Form of Quadratic Function:
y = ax2 + bx + c
Standard Form of Quadratic Equation:
0 = ax2 + bx + c
Solutions
A SOLUTION to a quadratic equation is a value for x,
that will make 0 = ax2 + bx + c true.
Note: A quadratic equation always have 2 solutions.
SOLVING BY FACTORING
Factoring
Solve by factoring;
2x2 – 11x = -15
Factoring
Solve by factoring;
x2 + 7x = 18
Solve by Factoring:
1. 2x2 + 4x = 6
2. 16x2 – 8x = 0
Solving by Finding Square Roots
For any real number x;
x2 = n
x=± n
Example: x2 = 25
Solve
Solve by finding the square root;
5x2 – 180 = 0
Solve
Solve by finding the square root;
4x2 – 25 = 0
Try Some!
Solve by finding the Square Root:
1. x2 – 25 = 0
2. x2 – 15 = 34
3. x2 – 14 = -10
4. (x – 4)2 = 25
Quadratic Equations
SOLVING BY GRAPHING
Solving by Graphing
For a quadratic function,
y = ax2 +bx + c
a zero of the function (where a function
crosses the x-axis) is a solution of the
equations
ax2 + bx + c = 0
Examples
Solve x2 – 5x + 2 = 0
Examples
Solve x2 + 6x + 4 = 0
Examples
Solve 3x2 + 5x – 12 = 8
Examples
Solve x2 = -2x + 7
Complex Numbers
Simplifying Radicals
If the number has a perfect square factor, you
can bring out the perfect square.
EX:
18
75
48
You Try!
27
200
75
Try this:
Solve the following quadratic equations
by finding the square root:
4x2 + 100 = 0
What happens?
Complex Numbers
Imaginary Number: i
The Imaginary number
i = -1
This can be used to find the root of any negative number.
r  i r
For example:
-9
-75
Properties of i
i = -1
i =
2
( -1)
2
= -1
i = i (i) = -1i
3
2
i = i (i ) = (-1)(-1) =1
4
2
2
Graphing Complex Number
Absolute Values
Absolute Values
Operations with Complex Numbers
The Imaginary unit, i, can be treated as a variable!
Adding Complex Numbers:
(8 + 3i) + ( -6 + 2i)
You Try!
1. 7 – (3 + 2i)
2. (4 – 6i) + 3i
Operations with Complex Numbers
Multiplying Complex Numbers:
Example: (5i)(-4i)
Example: (2 + 3i)(-3 + 5i)
Try Some!
1.
(6 – 5i)(4 – 3i)
2.
(4 – 9i)(4 + 3i)
Now we can SOLVE THIS!
Solve
4x2 + 100 = 0
Completing the Square
5 ways to solve
There are 5 ways to solve quadratic equations:
Factoring
Finding the Square Root
Graphing
Completing the Square
Quadratic Formula
Solving a Perfect Square Trinomial
We can solve a Perfect Square Trinomial using square roots.
A Perfect Square Trinomial is one with two of the same factors!
X2 + 10x + 25 = 36
Solving a Perfect Square Trinomial
x2 – 14x + 49 = 81
What if it’s not a Perfect Square Trinomials?!
If an equation is NOT a perfect square Trinomial,
we can use a method called
COMPLETING THE SQUARE.
Completing the Square
Using the formula for completing the square, turn
each trinomial into a perfect square trinomial.
Solving by Completing the Square
Solve by completing the square:
x2 + 6x + 8 = 0
Solving by Completing the Square
Solve by completing the square:
x2 – 12x + 5 = 0
Solving by Completing the Square
Solve by completing the square:
x2 – 8x + 36 = 0
Classwork: Solve by completing the square
1)
x  4 x  21
2)
x  8x  33
3)
x  10x  5
4)
x  5x  5  0
5)
x x70
2
2
2
2
2
Solving Quadratic Equations
Solve by Factoring
2x2 – x = 3
x2 + 6x + 8 = 0
Solve by Finding the Square Root
5x2 = 80
2x2 + 32 = 0
Solve by Graphing
X2 + 5x + 3 = 0
3x2 – 5x – 4 = 0
Solve by Completing the Square
X2 – 3x = 28
x2 + 6x – 41 = 0
5 ways to solve
There are 5 ways to solve quadratic equations:
Factoring
Finding the Square Root
Graphing
Completing the Square
Quadratic Formula
Quadratic Formula
The Quadratic Formulas is our final way to Solve!
It works when all else fails!
Examples
2x2 + 6x + 1 = 0
Examples
X2 – 4x + 3 = 0
3x2 + 2x – 1 = 0
X2 = 3x – 1
8x2 – 2x – 3 = 0
Discriminant
Discriminant
Discriminant
1.
IF the Discriminant is POSITIVE then
there are 2 REAL solutions
2.
IF the Discriminant is ZERO then there
is ONE REAL solution
3.
IF the Discriminant is NEGATIVE then
there are 2 IMAGINARY solutions.
Using the Discriminant
The weekly revenue for a company is: R = -3p2 + 60p
+ 1060, where p is the price of the company’s product.
Use the discriminant to find whether there is a price
the company can sell their product to reach a
maximum revenue of $1500?