unit 5 quadratic expressions and...

Unit 5 Quadratic Expressions and Equations

Test Date: __________________________

Name: ___________________________________________________________________________________

By the end of this unit, you will be able to…

Add, subtract, and multiply polynomials Solve equations involving the products of monomials and polynomials Find squares of sums and differences Find the product of a sum and a difference Factor polynomials Solve quadratic equations Factor binomials that are the difference of squares Use the difference of squares to solve equations Factor perfect square trinomials Solve equations involving perfect squares

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Table of Contents

Vocabulary Organizer ...................................................................................................................................................................... 4

5.1 Adding Polynomials .................................................................................................................................................................. 5

Standard Form ............................................................................................................................................................................... 5

5.1 Subtracting Polynomials ......................................................................................................................................................... 5

5.2 Multiplying Polynomials by Monomials ............................................................................................................................ 6

Method 1: Distributive Property ............................................................................................................................................ 6

Method 2: Box Method ............................................................................................................................................................... 6

5.2 Multiplying Polynomials by Polynomials ......................................................................................................................... 7

Method 1: Box Method ............................................................................................................................................................... 7

Method 2: Distributive Property ............................................................................................................................................ 7

5.3 Special Products ......................................................................................................................................................................... 8

Squares of Sums ............................................................................................................................................................................ 8

Squares of Differences ................................................................................................................................................................ 9

Product of a Sum and a Difference ......................................................................................................................................... 9

5.4 Factoring with a GCF ............................................................................................................................................................... 10

Reverse Box Method .................................................................................................................................................................. 10

Reverse Distributive Property ............................................................................................................................................... 11

Solving Equations by Factoring............................................................................................................................................. 12

5.5 Factoring Trinomials (leading coefficient 1) ................................................................................................................. 13

Diamond Problems and Factor Tables ............................................................................................................................... 13

Factoring ........................................................................................................................................................................................ 14

Solving Equations ....................................................................................................................................................................... 15

5.6 Factoring Trinomials (leading coefficient a) ................................................................................................................. 16

Slide and Divide ........................................................................................................................................................................... 16


5.7 Difference of Squares .............................................................................................................................................................. 19


5.8 Perfect Squares ......................................................................................................................................................................... 21

Identifying Perfect Square Trinomials ............................................................................................................................... 21

Solving Equations with Perfect Squares ............................................................................................................................ 22

Equations Using the Square Root Property ...................................................................................................................... 22

Interpreting Your Answer in Word Problems ................................................................................................................. 22

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Vocabulary Organizer Warm Up: With your group, think of as many words as you can that begin with the following prefixes: Mono- Bi- Tri-

Term

Polynomial

Monomial

Binomial

Trinomial

Perfect square trinomial

Degree

Quadratic

Leading Coefficient

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5.1 Adding Polynomials

Standard Form *Note: All answers need to be written in standard form—terms are in order from _______________________ ___________________________________________________________. Example 1: Write 3𝑥2 + 4𝑥5 − 7𝑥 in standard form.

Example 2: Write 5𝑦 − 9 − 2𝑦4 − 6𝑦3 in standard form.

Directions: Add like terms by grouping them.

1. (2𝑥2 + 5𝑥 − 7) + (3 − 4𝑥2 + 6𝑥)

2. (3𝑦 + 𝑦3 − 5) + (4𝑦2 − 4𝑦 + 2𝑦3 + 8)

5.1 Subtracting Polynomials Be Careful! Make sure to change ALL signs in the second polynomial!

1. (3 − 2𝑥 + 2𝑥2) − (4𝑥 − 5 + 3𝑥2) 2. (7𝑝 + 4𝑝3 − 8) − (3𝑝2 + 2 − 9𝑝)

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5.2 Multiplying Polynomials by Monomials Warm Up: Add or subtract the following polynomials.

1. (4𝑥2 + 3𝑥 + 12) − (3𝑥2 − 2𝑥 − 1) 2. (2𝑥2 + 7𝑥 + 2) + (𝑥2 − 3𝑥 − 6)

Method 1: Distributive Property

1. −3𝑥2(7𝑥2 − 𝑥 + 4)

2. 5𝑎2(−4𝑎2 + 2𝑎 − 7) 3. −6𝑑3(3𝑑4 − 2𝑑3 − 𝑑 + 9)

Method 2: Box Method

1. −3𝑥2(7𝑥2 − 𝑥 + 4)

2. 5𝑎2(−4𝑎2 + 2𝑎 − 7)

3. −6𝑑3(3𝑑4 − 2𝑑3 − 𝑑 + 9)

Solving Equations – Multiplying monomials by polynomials helps us solve complicated equations. Example: 2𝑥(𝑥 + 4) + 7 = (𝑥 + 8) + 2𝑥(𝑥 + 1) + 12

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5.2 Multiplying Polynomials by Polynomials

Method 1: Box Method

1. (𝑥 + 2)(𝑥 + 3) 2. (𝑥 − 4)(2𝑥 + 3)

3. (2𝑥 + 3)(𝑥 + 5) 4. (2𝑦2 + 3𝑦 − 1)(3𝑦2 − 5𝑦 − 2)

Method 2: Distributive Property

1. (𝑥 + 2)(𝑥 + 3) 2. (𝑥 − 4)(2𝑥 + 3)

3. (2𝑦 − 7)(3𝑦 + 5) 4. (6𝑥 + 5)(2𝑥2 − 3𝑥 − 5)

Application: A contractor is building a deck around a rectangular swimming pool. The homeowner would like one side of the pool to be 6 feet longer than the other. Write an expression for the area of the pool.

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5.3 Special Products Do Now: Create a square out of algebra tiles. (Remember: All sides should be the same length in a square) What is the side length of your square? _______________ What is the formula for the area of your square? _______________________ What is the area of your square? (Multiply polynomials) _______________ *You may wish to sketch your square at right*

Squares of Sums Write down the side length, area formula, and area of at least three other squares. Square #1 Square #2 Square #3 Side length:

Side length:

Side length:

Area formula:

Area formula:

Area formula:

Area:

Area:

Area:

What do you notice? Summary: Examples:

1. (𝑥 + 3)2 2. (3𝑥 + 5)2

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Squares of Differences What happens when some of our tiles are negative?

Swap all the 𝑥 tiles in your square for red – 𝑥 tiles. How does this affect the area? Summary: Examples:

1. (6𝑝 − 1)2 2. (𝑥 − 2)2

Product of a Sum and a Difference

What happens when we only switch to negative tiles on one side? Change the red – 𝑥 tiles on one side back into positive 𝑥 tiles. How does this affect the area? Summary: Examples:

1. (𝑥 + 4)(𝑥 − 4) 2. (3𝑥 + 2)(3𝑥 − 2)

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5.4 Factoring with a GCF Remember: GCF means “greatest common factor.” Warm Up:

1. List all the factors of 48.

2. What is the GCF of 9 and 12? 3. What is the GCF of 65 and 39?

4. What is the GCF of 𝑥3 and 𝑥2? 5. What is the GCF of 4𝑥5 and 6𝑥3?

Reverse Box Method Examples:

1. 27𝑥 − 9 2. 2𝑥2 + 5𝑥

3. −9𝑚4 − 18 4. 2𝑥5 + 3𝑥2 − 4𝑥

5. 48𝑢5 − 72𝑢4 + 36𝑢3

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Reverse Distributive Property Examples:

1. 27𝑦2 + 18𝑦

2. −4𝑎2𝑏 − 8𝑎𝑏2 + 2𝑎𝑏

3. 15𝑤 − 3𝑣

4. 7𝑢2𝑡2 + 21𝑢𝑡2 − 𝑢𝑡 5. 2𝑥2𝑦2 + 5𝑥𝑦3

Steps:

1. Find the ________________.

2. Take it ________________

_________________________.

3. What is left?

___________________________

__________________________.

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Solving Equations by Factoring The Zero Product Property In words: If the product of two factors is 0, then at least one of the factors must be _________. In symbols: If 𝑎 ∗ 𝑏 = 0, then _______________________________________________________________________. Examples (Already Factored):

1. Solve (𝑥 − 4)(𝑥 + 7) = 0. 2. Solve (2𝑑 + 6)(3𝑑 − 15) = 0.

3. Solve 3𝑛(𝑛 + 2) = 0. 4. Solve (𝑥 − 2)(4𝑥 + 1) = 0. Examples (You need to do the factoring):

1. Solve 𝑐2 = 3𝑐. 2. Solve 8𝑏2 = 40𝑏.

3. Solve 𝑥2 = −10𝑥. 4. Solve 3𝑘2 = 24𝑘.

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5.5 Factoring Trinomials (leading coefficient 1) Warm Up: List all the factors of the following numbers: 27 13 4 36

Diamond Problems and Factor Tables The rules of diamond problems:

You must find 2 numbers to put on the left and right sides of the X.

When you multiply your two numbers together, they must equal the top number.

When you add your two numbers together, they must equal the bottom number. Good luck!

1. 2. 3. 4. 5. 6.

Tips and Tricks:

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Factoring

1. Draw a diamond problem. For 𝑎𝑥2 + 𝑏𝑥 + 𝑐, write 𝑎 ∙ 𝑐 at the top and 𝑏 at the bottom.

2. Find two numbers that multiply to equal _______, and add to equal _______.

a. If you can’t find them, make a table of all the factors of _________.

3. Write your answer in factored form: (𝑥 + _____)(𝑥 + _____ ). Examples:

1. Factor 𝑥2 + 9𝑥 + 20.

2. Factor 𝑥2 + 11𝑥 + 24.

3. Factor 𝑥2 + 15𝑥 + 36.

4. Factor 𝑥2 + 10𝑥 + 9.

5. Factor 𝑥2 − 8𝑥 + 12.

6. Factor 𝑥2 − 11𝑥 + 28.

7. Factor 𝑥2 + 2𝑥 − 15.

8. Factor 𝑥2 − 7𝑥 − 18.

Tip!: You can always check your answer by multiplying your factors together!

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Solving Equations

1. Make sure equation is equal to _______.

2. _________________.

3. Set each factor equal to _______.

4. _____________ each equation.

Examples:

1. Solve 𝑥2 + 3𝑥 − 18 = 0.

2. Solve 𝑥2 − 15𝑥 + 36 = 0.

3. Solve 𝑥2 − 3𝑥 + 2 = 0.

4. Solve 𝑥2 − 3𝑥 = 70.

5. Solve 𝑥2 + 12𝑥 = −32.

6. Solve 𝑥2 − 𝑥 − 72 = 0.

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5.6 Factoring Trinomials (leading coefficient a)

Slide and Divide

1. If possible, factor out a _________.

2. Draw a diamond problem. For 𝑎𝑥2 + 𝑏𝑥 + 𝑐, write 𝑎 ∙ 𝑐 at the top and 𝑏 at the bottom.

3. Find two numbers that multiply to equal _______, and add to equal _______.

a. If you can’t find them, make a table of all the factors of _________.

4. Write your answer in factored form: (𝑥 + _____)(𝑥 + _____ ). Divide by ________!

5. If possible, reduce/simplify any fractions.

6. Kick any denominators to the front of the factor. Examples:

1. Factor 3𝑥2 − 17𝑥 + 20.

2. Factor 12𝑥2 + 11𝑥 − 5.

3. Factor 5𝑥2 + 27𝑥 + 10.

4. Factor 6𝑥2 + 17𝑥 + 5.

5. Factor 2𝑥2 + 3𝑥 − 5.

6. Factor 14𝑥2 − 11𝑥 + 2.

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7. Factor 3𝑥2 − 11𝑥 − 20.

8. Factor 6𝑥2 − 𝑥 − 15.

Don’t forget step 1! Whenever there is a GCF, factor it out! Slide and divide doesn’t work if you don’t! Examples:

1. Factor 12𝑥2 − 69𝑥 + 45.

2. Factor 5𝑥2 + 20𝑥 − 105.

3. Factor 8𝑥2 − 4𝑥 − 4.

Challenge Problems

1. Six times the square of a number 𝑥 plus 11 times the number equals 2. What are the possible values for 𝑥?

2. Factor −4𝑥2 − 15𝑥 − 14.

3. Factor 4𝑥2 − 25.

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Solving Equations Factor, then use the ZPP! Examples:

1. Solve 2𝑥2 + 9𝑥 − 18 = 0. 2. Solve 7𝑥2 + 3 = 10𝑥.

3. Solve 2𝑥2 − 13𝑥 = −15. 4. Solve 3𝑥2 − 5𝑥 = 2.

5. Alex throws the discus at a school meet. The equation ℎ = −16𝑡2 + 38𝑡 + 5 models his throw. After how many seconds does the discus hit the ground?

6. Hannah dives from a 36-foot platform. The equation ℎ = −16𝑡2 + 14𝑡 + 36 models the dive. How long will it take Hannah to reach the water?

Shortcut:

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5.7 Difference of Squares Think Back: What is our shortcut for multiplying binomials of the form (𝑎 + 𝑏)(𝑎 − 𝑏)? Warm Up: Multiply:

1. (𝑥 + 5)(𝑥 − 5) = _____________________. 2. (2𝑥 + 3)(2𝑥 − 3) = _______________________. How can we use this trick to help us when factoring polynomials of the form 𝑎2 − 𝑏2? Summary: Examples: Factor each polynomial.

1. 𝑥2 − 64

2. 𝑥2 − 26

3. 16ℎ2 − 9𝑎2

4. 121 − 4𝑏2

5. 81 − 𝑐2

6. 64𝑔2 − ℎ2

Tip!: Don’t forget to factor out a GCF whenever you can. Tip #2!: We can use this technique more than once, or mix this technique with other techniques. Examples: Factor each polynomial.

7. 27𝑔3 − 3𝑔

8. 9𝑥3 − 4𝑥

9. −4𝑦3 + 9𝑦

10. 𝑏4 − 16

11. 625 − 𝑥4

12. 5𝑥5 − 45𝑥

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Solving Equations Factor as much as you can, then use ZPP. Examples:

1. Solve 9𝑚2 − 144 = 0.

2. Solve 18𝑥3 = 50𝑥.

3. During an accident, skid marks may result from sudden braking. The formula 1

24𝑠2 = 𝑑

approximates a vehicle’s speed 𝑠 in miles per hour given the length 𝑑 in feet of the skid mark on dry concrete. If a skid mark is 54 feet long, how fast was the car traveling when the brakes were applied?

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5.8 Perfect Squares Think Back: What is our shortcut for multiplying binomials of the form (𝑎 + 𝑏)2 and (𝑎 − 𝑏)2? Warm Up: Multiply:

1. (𝑥 + 5)(𝑥 + 5) = _____________________. 2. (2𝑥 − 3)(2𝑥 − 3) = _______________________. How can we use this trick to help us when factoring polynomials of the form 𝑎2 + 2𝑎𝑏 + 𝑏2? Summary:

Identifying Perfect Square Trinomials Before we can use this shortcut to help us factor, we need to know how to recognize perfect square trinomials. We know that they must fit the form 𝑎2 + 2𝑎𝑏 + 𝑏2. Questions to ask yourself:

1. Is the first term a perfect square? 2. Is the last term a perfect square? 3. Take the square root of the 1st and 3rd terms to find a and b. Is the middle term equal to 2ab?

Examples: Determine whether each trinomial is a perfect square. Write yes or no. If so, factor it.

1. 4𝑦2 + 12𝑦 + 9 2. 9𝑥2 − 6𝑥 + 4

3. 9𝑦2 + 24𝑦 + 16 4. 2𝑎2 + 10𝑎 + 25

5. 25𝑥2 − 30𝑥 + 9 6. 49𝑦2 + 42𝑦 + 36

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Solving Equations with Perfect Squares

1. Factor. 2. Write perfect squares as repeated factors. 3. Use ZPP to solve!

Examples: Solve each equation.

1. 9𝑥2 − 48𝑥 = −64 2. 𝑎2 + 12𝑎 + 36 = 0

Equations Using the Square Root Property Square Root Property: To solve a quadratic equation with the form 𝑥2 = 𝑛, take the _________________________ of each side. Examples:

1. 𝑥2 = 25 2. (𝑦 − 6)2 = 81

3. (𝑥 + 6)2 = 12

Interpreting Your Answer in Word Problems Some word problems should only have one answer. You must examine the problem to determine which answer is best. Example: During an experiment, a ball is dropped from a height of 205 feet. The formula ℎ = −16𝑡2 + ℎ0 can be used to approximate the number of seconds (t) it takes for the ball to reach height (h) from an initial height (ℎ0). Find the time it takes the ball to reach the ground.

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