1

Solving Quadratics by Factoring (Day 10–1)

Recall: Factor 822 xx completely.

What values of x make 0)2)(4( xx true?

Graph 822 xxy .

What is the connection?

2

Quadratic Equation (Standard Form):

1. Find the roots: 64)( 2 xxf 2. Find the zeroes: xx 355 2

3. Solve: 6022 2 xx 4. Find the zeroes: xx 1235 2

5. Find the x-intercepts: 26)( 2 xxxg

Steps for Solving a Quadratic Equation by Factoring

Write equation in standard form.

Factor the quadratic equation. (GCF, D2PS, X-Box, Grouping)

After the problem has been factored we will complete a step called the “T-chart”.

Create a T-chart separating the two ( ).

Once ( ) are separated, set each ( ) = to 0 and solve for the variable.

If necessary, check each of the roots in the ORIGINAL quadratic equation.

3

Classwork 10–1

1. Solve for the roots: 020122 xx 2. Find the x-intercepts: 19)( 2 xxh

3. Find the zeroes: xx 62 4. Find the zeroes: 5210 2 xx

5. Solve for the zeroes of 1072 xxy . Then, graph the equation.

What is the connection between the zeroes

and what you see on the graph?

4

6. Tony makes a phone call at a pay phone. The charge is $0.25 for placing the call and

$0.10 for each minute. Tony has $2.10 in change in his pocket. Write an inequality that

can be used to find m, the maximum number of minutes that Tony can talk on the phone.

Solve this inequality algebraically to find the maximum number of whole minutes he can

talk on the phone.

7. Graph the inequalities and label the solution set with an “S.”

4

3

2

73

xy

yx

y

x

5

Solving Quadratics by Completing the Square (Day 10–2)

Why is 1682 xx an example of a perfect square trinomial? Hint: Factor it.

SOLVING A QUADRATIC EQUATION BY

COMPLETING THE SQUARE

014122 2 xx

1. Rearrange the equation:

Get terms with variables on the left hand side.

Get c# (constant) by itself on the right hand side.

2. If a# > 1 then divide through by a#.

3. Complete the Square

Identify the b#.

Take half of b - square it - add it to both sides)

(This will form a perfect square trinomial).

Write expression as a perfect square trinomial.

Simplify the # on the right side of the = sign.

4. Solve for x.

Square root both sides

Put a on the right in front of term

Solve for x to find the roots.

6

Solve by completing the square. Express each root in simplest radical form when necessary.

1. 682 xx 2. 02463 2 xx

3. xx 2482 2

4. If xx 622 is solved by completing the square, an intermediate step would be:

(1) 7)3( 2 x (3) 11)3( 2 x

(2) 7)3( 2 x (4) 34)6( 2 x

7

Classwork 10–2

Solve by completing the square. Express each root in simplest radical form when necessary.

1. 24123 2 xx

2. xx 4622 2

8

For questions #3 & 4:

a) Write a formula for the given sequence.

b) Use the formula to find 10a .

3. 7, 14, 28, 56, 112, … 4. 1, 1.5, 2, 2.5, 3, …

a) a)

b) b)

5. Which equation is an example of the use of the associative property of addition?

(1) xx 77 (3) )3(3)( yxyx

(2) yxyx 33)(3 (4) 3)()(3 yxyx

6. Solve and graph the following inequality. Then express your solution in interval notation.

166)4(2 xx

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More Completing the Square (Day 10–3)

1. Milton made a mistake when beginning to solving 0482 2 xx by completing the

square. Explain and correct the mistake, and then finish solving the equation.

482 2 xx

xx 82 2 +16 = 4 +16

201682 2 xx

2. Solve by completing the square and, if necessary, express result in simplest radical form:

11244 2 xx

10

Solve each of the following by completing the square and, if necessary, express result in

simplest radical form:

3. 1222

1 2 xx 4. 9)4(4 xx

5. Why is complete the square a difficult method to solve this equation?

832 xx

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Classwork 10–3

Solve each of the following by completing the square. Express your answer in simplest

radical form when necessary.

1. 057183 2 xx 2. 1242 xx

3. Solve for V in terms of P and R: R

VP

2

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4. Which property is illustrated in the following equation?

)6()6(8)8)(6( xxxxx

(1) Distributive property

(2) Associative property of addition

(3) Associative property of multiplication

(4) Commutative property of multiplication

5. Which equation represents the line that passes through the points (–1, –2) and (3, 10)?

(1) 13 xy (3) 24 xy

(2) 13 xy (4) 24 xy

6. Write a recursive and explicit equation for the sequence 7, 21, 63, 189, …

7. An online music club has a one-time registration fee of $13.95 and charges $0.49 to buy

each song. If Emma has $50.00 to join the club and buy songs, what is the maximum

number of songs she can buy?

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The Quadratic Formula (Day 10–4)

Completing the square and factoring are not always the best method to use when solving a

quadratic equation.

Why are completing the square and factoring not good options for the quadratic below?

04127 2 pp

To remember formula sing/hum the phase below to the “pop goes the weasel song”

“x =’s negative b, plus or minus the square root of b2 minus 4 a c, all over 2 a”

******WRITE THE FORMULA DOWN AS YOU SING THE SONG******

Quadratic Formula:

Steps for using the Quadratic Formula

Get equation equal to ZERO!!!

Put equation in standard form: __________________________

Identify the a, b, and c #’s.

Plug into the formula and simplify.

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Solve for the roots/zeroes of each of the following quadratic equations using the quadratic

formula. If necessary, express your answers in simplest radical form.

1. xx 9182 2 2. 35)( 2 xxxh

3. 20122 xx 4. 142 2 pp

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Classwork 10–4

Solve for the roots/zeroes of each of the following quadratic equations using the quadratic

formula. If necessary, express your answers in simplest radical form.

1. 1222 xx 2. 782)( 2 xxxk

3. The perimeter of a triangle can be represented by the expression 8105 2 xx . Write a

polynomial that represents the measure of the third side.

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4. Paula just bought a new car for $18,600. She looked up on the internet that her model is

expected to depreciate in value by 18.5% every year. If she plans on owning the car for 3

years, what should she expect the value to be after the 3 years?

5. Jack Eichel and Ryan O’Reilly are hungry for pizza and breadsticks. They order food from

the same restaurant at the same prices. Jack orders three large pizzas and two orders of

breadsticks and pays $44. Ryan orders five large pizzas and four orders of breadsticks for

$76. How much do a large pizza and an order of breadsticks cost?

6. Given the equation 0)2)(7( xx , what is the smaller root?

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More with the Quadratic Formula (Day 10–5)

Quadratic Formula:

Solve for the roots/zeroes of the following quadratic equations using the quadratic formula. If

necessary, get your answer in simplest radical form.

1. 1283 2 xx 2. 235)( 2 xxxf

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Solve each of the following using the most appropriate method.

3. Find the roots: 01255 2 x 4. Find the x-intercepts: xxxf 22)(

5. Which method is best to solve the equation 572 xx ? Why?

(1) Complete the square since the “b” value is positive.

(2) Complete the square since the “b” value is even.

(3) X-box factoring since it can factor.

(4) Quadratic formula – other techniques failed or are very difficult.

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Classwork 10–5

Solve for the roots/zeroes of the following quadratic equations using the quadratic formula. If

necessary, get your answer in simplest radical form.

083.1 2 xx bb 482.2 2

3. Matt made a mistake when solving 0252 2 xx by the quadratic formula. Explain and

correct the mistake.

8,2

35

95

16255

)2()2(4)5()5( 2

x

x

x

x

x

20

4. Simplify the following and expressing all answers with only positive exponents.

a) )9)(3( 4255 zxyyx b) )2(4)72(5 2334 yyyyyy

c) 2)63( x d) ab

baab 22 43

5. Solve the system of equations graphically.

1

832

yx

xy

y

x

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Mixed Quadratic Problems (Day 10–6)

METHOD REASON TO USE THIS METHOD QUICK PROCEDURE

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1. A rocket carrying fireworks is launched from a hill 80 feet above a lake. The rocket will fall

into the lake after exploding at its maximum height. The rocket’s height above the

surface of the lake is given by 806416)( 2 ttth , where t is the time in seconds and h is

the height in feet.

a) What does t represent? b) What does )(th represent?

c) What is the input of the function? d) What is the output of the function?

e) How long will it take for the rocket to reach 128 feet?

f) Find the height at 0 seconds. What is the graphic name of this point?

g) Find the height at 2 seconds.

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2. The area of a rectangular playground enclosure at Happy Times Nursery School is 600 sq.

meters. The length is 25 meters longer than the width. Find the dimensions of the

playground.

3. Collin is building a deck on the back of his house. He has enough lumber for the deck to

be 144 square feet. The length should be 10 feet more than its width. What should the

dimensions of the deck be?

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Classwork 10–6

1. After t seconds, a ball tossed in the air from the ground level reaches a height of h feet

given by the equation ttth 14416)( 2 .

a) What is the height of the ball at 4 seconds?

b) After thrown, when will the ball hit the ground?

2. The length and width of a rectangle are consecutive odd integers. If the area of the

rectangle is 63 in2, find the dimensions of the rectangle. Only an algebraic solution will be

accepted.

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3. Consider the quadratic equation xx 1053 2 .

a) State two possible methods to solve for the roots.

b) Using one of the ways stated above, solve the quadratic equation xx 1053 2 in

simplest radical form.

4. Simplify:

a) 802 b) 62 1253 xx

5. Factor:

a) 814 2 x b) 5223 1014 yxyx

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Exam Quadratic Problems (Day 10–7)

1. The Demon Drop at Cedar Point in Ohio takes riders to the top of a tower and drops them

60 feet. A function that approximates this ride is 606416 2 tth , where h is the height

of the riders in feet and t is the time in seconds. To the nearest tenth, how many seconds

does it take for riders to hit the ground?

2. Consider the quadratic equation 483)12)(2()( 2 xxxxw .

a) Simplify )(xw and write it as a trinomial.

b) Solve for x when 0)( xw .

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3. Tammy found the zeroes of the function )(xf to be –3 and 8. Write a quadratic equation

that could represent Tammy’s function )(xf .

4. Milton made a mistake when solving 0362 xx by the quadratic formula. Circle,

explain and correct the mistake.

623

2

626

2

246

2

12366

)1(2

)3()1(4)6()6( 2

x

x

x

x

x

28

Classwork 10–7

1. Emma made a mistake when solving 0822 xx by completing the square. Explain

and correct the mistake.

822 xx

xx 2( 2 +1 ) = 8 +1

2

31

9)1( 2

x

x

x

2. Solve the equation for y: 124)3( 2 yy

3. The roots of a function are 2x and 9x . Write a possible quadratic function for these

roots.

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4. The senior class at Bay High School buys jerseys to wear to the football games. The cost

of the jerseys can be modeled by the equation 254.21.0)( 2 xxxC , where )(xC is the

amount it costs to buy x jerseys. How many jerseys can they purchase for $500?

5. Using the given graph, state the inequality that has

the negative slope.

6. A town’s population is currently at 100,000 and is growing at an annual rate of 5%. What

will the population be in 8 years?

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More Exam Quadratic Problems (Day 10–8)

1. Find the zeroes of 49)3()( 2 xxf algebraically.

2. The height, H, in feet of an object dropped from the top of a building after t seconds is

given by 14416)( 2 ttH .

a) Determine the height of the building.

b) How many feet did the object fall between one and two seconds after it was

dropped?

c) Determine, algebraically, how many seconds it will take for the object to reach the

ground.

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3. Amy solved the quadratic equation 04252 2 xx . She stated that the solutions to the

equations were 2

7 and –6. Do you agree with Amy’s solutions? Explain why or why not.

4. The length and width of a rectangle are consecutive even integers. If the area of the

rectangle is 224 in2, find the dimensions of the rectangle. Only an algebraic solution will

be accepted.

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Classwork 10–8

1. The solution of the equation 7)3( 2 x is:

(1) 73 (3) 37

(2) 73 (4) 37

2. Keith determines the zeroes of the function )(xf to be 6 and –5. Which of the following

could be Keith’s function?

(1) )6)(5()( xxxf (3) )6)(5()( xxxf

(2) )6)(5()( xxxf (4) )6)(5()( xxxf

3. When solving the equation 0782 xx by completing the square, which equation is a

step in the process?

(1) 9)4( 2 x (3) 23)4( 2 x

(2) 9)8( 2 x (4) 23)8( 2 x

4. Rhiannon was asked to solve this word problem: “The product of two consecutive even

integers is 224. What are the integers?” What type of equation should she create to solve

this problem?

(1) linear (3) quadratic

(2) exponential (4) absolute value

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5. Using the diagram below, write a formula to represent the number of blocks in the thn

diagram.

6. Tabitha solved 0822 xx by the quadratic formula below. Is her solution correct or

incorrect? Explain your reasoning.

7. Solve for x: 12

1

6

5

4

x

4,2

2

62

2

362

2

3242

)1(2

)8()1(4)2()2( 2

x

x

x

x

x