FACTORING QUADRATIC EXPRESSION
Algebra 2: Unit 5 Continued
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)
5. (x + 2) and (x + 4) 6. (x – 4) and (x – 5)
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!
Question
2nd sign
+Same
-Different
1st sign
+ - + or -
Answer(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
More Factoring!
When a does NOT equal 1.Steps
1. Slide2. Factor3. Divide4. Reduce5. Slide
Example!
Factor:1. 3x2 – 16x + 5
Example!
Factor:2. 2x2 + 11x + 12
Example!
Factor:3. 2x2 + 7x – 9
Try Some!
Factor1. 5t2 + 28t + 32 2. 2m2 – 11m + 15
March 20th Warm Up
Find the Vertex, Axis of Symmetry, X-intercept, and Y-intercept for each:1. y = x2 + 8x + 9
2. y= 2(x – 3)2 + 5
Quadratic Equations
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.
A quadratic equation always have 2 solutions.
5 ways to solve
There are 5 ways to solve quadratic equations:
FactoringFinding the Square RootGraphingCompleting the SquareQuadratic Formula
SOLVING BY FACTORING
Factoring
Solve by factoring;2x2 – 11x = -15
Factoring
Solve by factoring;x2 + 7x = 18
Factoring
Solve by factoring;1. 2x2 + 4x = 6 2. 16x2 – 8x = 0
3. x2 – 9x + 18 = 0
Solving by Finding Square Roots
For any real number x;X2 = n
x =
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
SOLVING BY GRAPHING
Quadratic Equations
Warm Up March 21st
A model for a company’s revenue is R = -15p2 + 300p + 12,000 where p is the price in dollars of the company’s product. What PRICE will maximize the Revenue?What is the maximum revenue?
Convert to vertex form: y = 2x2 + 6x - 8
5 ways to solve
There are 5 ways to solve quadratic equations:
FactoringFinding the Square RootGraphingCompleting the SquareQuadratic Formula
Solving by Graphing
For a quadratic function, y = ax2 +bx + c, a zero of the function, or 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
Quick Review
Simplifying RadicalsIf the number has a perfect square factor, you can bring out the perfect square.
EX:
Try Some
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
This can be used to find the root of any negative number.
EX
Properties of i
This pattern repeats!!
Graphing Complex Number
Absolute Values
Operations with Complex Numbers
The Imaginary unit, i, can be treated as a variableAdding Complex NumberEX: (8 + 3i) + ( -6 + 2i)
Try Some!
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
Absolute Values
Completing the Square
Warm Up
Factor each Expressions
5 ways to solve
There are 5 ways to solve quadratic equations:
FactoringFinding the Square RootGraphingCompleting the SquareQuadratic 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 Competing the Square
Solve by completing the square:X2 + 6x + 8 = 0
Solving by Competing the Square
Solve by completing the square:X2 – 12x + 5 = 0
Solving by Competing the Square
Solve by completing the square:X2 – 8x + 36 = 0
Solving Quadratic Equations
Warm Up
1. Write in Vertex Form: y = 2x2 + 6x – 8
2. Simplify |2i + 4|
3. Simplify (3i – 2)(5i + 3)
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:
FactoringFinding the Square RootGraphingCompleting the SquareQuadratic 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
IF the Discriminant is POSITIVE then there are 2 REAL solutions
IF the Discriminant is ZERO then there is ONE REAL solution
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?
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