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) 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: 9x2 +3x – 18 7p2 + 21 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 1st sign + or - (x+ )(x+ ) 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
Slide Factor Divide Reduce More Factoring! When a does NOT equal 1. Steps Slide Factor Divide Reduce
Example! Factor: 1. 3x2 – 16x + 5
Example! Factor: 2. 2x2 + 11x + 12
Example! Factor: 3. 2x2 + 7x – 9
Try Some! Factor 1. 5t2 + 28t + 32 2. 2m2 – 11m + 15
March 20th Warm Up Find the Vertex, Axis of Symmetry, X-intercept, and Y-intercept for each: y = x2 + 8x + 9 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: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula
SOLVING BY FACTORING
Solve by factoring; 2x2 – 11x = -15
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
Quadratic Equations Solving by Graphing
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: Factoring Finding the Square Root Graphing Completing the Square Quadratic 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 Radicals If 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 variable Adding Complex Number EX: (8 + 3i) + ( -6 + 2i)
Try Some! 7 – (3 + 2i) (4 – 6i) + 3i
Operations with Complex Numbers Multiplying Complex Numbers: Example: (5i)(-4i) Example: (2 + 3i)(-3 + 5i)
Try Some! (6 – 5i)(4 – 3i) (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: 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 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 Write in Vertex Form: y = 2x2 + 6x – 8 Simplify |2i + 4| 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: 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 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?