factoring quadratic expression algebra 2: unit 5 continued

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



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



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


Solve by completing the square:X2 – 12x + 5 = 0


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



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

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?

factoring quadratic expression algebra 2: unit 5 continued

Documents