pap algebra 2 unit 4b - hhs algebra ii -...

PAP Algebra 2

Unit 4B Quadratics (Part 2)

Name________________________

Period______

1

After Test WS: 4.6 Solve by Factoring Name_________________________________

PAP Algebra 2

Factor.

1. 2

6 8x x+ + 2. 2

4 8 3x x+ + 3. 2

2 7 5x x+ +

4. 2

2 3 1x x+ + 5. 2

7 6x x+ + 6. 2

3 10 3x x+ +

7. 2

12 11 2x x+ + 8. 2

8 10 3x x− + 9. 2

3 17 6x x+ −

Solve by factoring.

11. 2

20 12 3x x= + 12. 2

10 11 3x x+ = − 13. 2

25 2 15x x− = −

3

14. 2

40 12 63x x+ = 15. 3 2

2x x x= + 16. 2

11 5 2x x= +

17. 3 2

6 15 6 0x x x− + = 18. 2

6 28 16x x+ = − 19. 2

5 6x x− = +

4

of 2

Solving Quadratics by Factoring Name_________________________________Per_________

Worksheet 4.6

Solve by factoring.

1. 25 6 0+ + =x x 2. 2

12 0− − =x x 3. 29 18 0− + =a a

4. 22 19 5+ − =t t 5. 2

15 30 6+ + = −x x 6. 210 16+ = −d d

7. 22 6 4 0+ + =x x 8. 2

3 12 15− =a a 9. 26 9 0− + =c c

10. 25 14 8 0− + =x x 11. 2

7 9− =h 12. 27 15 6 4− + =t t

13. 210 18 7+ + = −d d 14. 2

4 46 3− =x 15. 211 32 17 20− + =a a

16. 24 12 9 0+ + =n n 17. 2

5 11 3 2 3− − = +x x x 18. 26 15 36 0− − =t t

Find the dimensions of the rectangles below.

19. 20.

520 in2

x +7

2x

234 ft2

x +8

x+3

5

of 2

Write a quadratic equation to represent each situation. Solve by factoring.

1. Thirty-five less than the square of a number is the same as 2 times the number.

Find the number.

2. Find two consecutive positive integers whose product is 72.

3. Find two consecutive odd integers such that the square of the first, added to 3 times the

second, is 34.

4. Find three consecutive integers such that the square of the first, increased by the last, is 32.

5. Find three consecutive integers such that the product of the first and third exceeds the product of 8

and the second by 32.

6. Ms. Powers can kick a soccer ball up into the air with an initial velocity of 44 feet per second. Using

the formula h = vt – 16t2, where h = height above the ground, v = initial velocity, and

t = time in seconds, find how many seconds it will take for the ball to return to the ground.

7. Suppose you transform a square by increasing one side by 4 units and decreasing the other side by 2

units. If the area of the resulting rectangle equals 16, how many units long was the side of the original

square?

8. Suppose you transform a square by increasing one side by 12 units and decreasing the other side by 8

units. If the area of the resulting rectangle equals 44 how many units long was the side of the original

square?

9. The length of a rectangle is 4 more than the width. Find the width of the rectangle if the area is 32.

10. The length of a rectangle is 5 times the width. If the length and width are both increased by 2, the new

area would be 85. What is the original area?

6

PAP Algebra II

4.6B Notes

GCF – greatest common factor

EX1) 6 9 3xy x y− + EX2) 2 32 10a b ab−

EX3) 4 5 3 2 26 8 12r m r m r m− + EX4) 4 3

3 4z y+

Difference of Squares ( )( )2 2a b a b a b− = − +

EX 5) 281x − EX 6) 2

49y + EX 7) 24 25x −

EX8) 2 29 49a b− EX9) 4 6

16m r− EX10) 1081c x−

Sum/Difference of Cubes 3 3 2 2( )( )a b a b a ab b+ = + − + 3 3 2 2

( )( )a b a b a ab b− = − + +

EX 5) 327x − EX 6) 3

64y + EX 7) 38 1x −

EX8) 3 6216c x− EX9) 6 3

8m r− EX10) 3 327 125a b−

Factor by Grouping

EX 11) 3 26 5 30x x x+ + + EX1 2) 3 2

6 3 4 2x x x− − +

7

COMBINED….

EX 14) 3 26 27 30d d d− + EX 15) 3 2

6 8 8y y y− − +

EX 16) 325g g− EX 17) 3

4 500c− +

EX 18) 4 25 20a a− EX 19) 3 2

5 9 45v v v− − +

Ex 20) Find three consecutive integers such that the product of the first and the third is 35 greater

than the product of the second and 5.

EX 21) Find four consecutive even integers such that the product of the second and fourth is 16 less

than the product of -3 and the sum of the first and third.

8

PAP Algebra II

WS 4.6B Name: ______________________________________________

Factor the following completely. If it does not factor state “not factorable”.

__________________________ 1) 24 12 28x x+ − __________________________ 2) xxx 12202

23+−

__________________________ 3) 3 364 27m z+ __________________________ 4) 3 2

2 3 6x x x+ − −

__________________________ 5) 39 16x x− __________________________ 6) 2

2 11 12m m− +

__________________________ 7) 3 26 4 24f f f+ − − __________________________ 8) 2 7

9 25p q q−

__________________________ 9) 38 125x − __________________________ 10) 3 2

20 4 5 1n n n+ − −

9

11) Find two consecutive odd integers such that the square of the larger diminished by

three times the smaller equals 34.

12) One number is 10 less than 8 times another number. If their product is -3, find the

numbers.

13) If the sides of a square garden are increased by 4 meters, the area of the new larger

square garden is 56 square meters more than the area of the original smaller garden.

How long was the side of the original smaller garden?

14) One leg of a right triangle is 3 cm shorter than the other leg. If the hypotenuse is

15 cm, what is the length of the longer leg?

15) A path of uniform width is constructed around a 20 X 25 meter rectangular garden.

If 196 square meters of brick are used to construct the path, what is the width of

the path?

10

PAP Algebra II

Notes 4.7 Complex Roots, Solve by Graphing, Solve by Quadratic Formula

Complex numbers

i43 + 10i 11−

Real Numbers 5 736.2 e π 7

3−

Rational Numbers

Irrational Numbers can be written as a fraction can not be written as a fraction 8

1− 25. 2.−

2 e π 11 7 Integers

5− 439 0 12

Today we are going to work with some imaginary (i.e. complex) numbers! ☺

The base for an imaginary number is i , which is the square root of negative 1.

1i = − This is imaginary because it does not exist. Put 1− in your calculator and

what does it say?

Complex numbers have a real part, and an imaginary part. a bi+ is the form of a complex number. “a” is the real part,

and “b” is the imaginary part.

EX1) Identify the real and imaginary part of these complex numbers:

a) 3 2i+ real: ______ imaginary: ______

b) 7 9i− real: ______ imaginary: ______

c) 5 3i − real: ______ imaginary: ______

d) 4i real: ______ imaginary: ______

e) 8 real: ______ imaginary: ______

This applies to quadratics because sometimes we will have roots that are not real.

11

An equation of the form ax2 + bx + c = 0 can be solved by using the quadratic formula: a

acbbx

2

42

−±−=

Use this formula to solve each of the following. Remember to set equal to zero first!

1. 8x2 – 2x - 1 = 0 2. 2x2 + 3x - 1 = 0

3. x2 – 2x + 1 = 0 4. x2 – 4x + 13 = 0

5. -x2 + 4x = 5 6. -5x2 +7x - 4 = 2x +3

In the quadratic formula, the expression acb 42

− is called the discriminant. We can use this to

determine how many solutions an equation has and if they are real or imaginary.

Value of

discriminant acb 4

2− >0 acb 4

2− =0 acb 4

2− <0

Number and type of

solutions

Example Graph

Find the discriminant of the following equations and give the number and type of solutions of the

equation. Remember to set equal to zero!

1. 05272

=+− xx 2. 0121232

=++ xx 3. xxx 7615322

−=+−

12

PAP Algebra II

WS 4.7 Name: ___________________________

Use the quadratic formula to find the roots for the following quadratics. Tell whether the roots are

real or imaginary. Put all complex roots in the form a+bi.

1) 25 3y x x= + + 2) 352

2−−= xxy 3) 2

3 2y x x= − − −

Roots: ___________ Roots: ___________ Roots: ___________ real imaginary real imaginary real imaginary

4) 22 5 3y x x= − − + 5) 2

2 5 7y x x= − − 6) 27y x x= − +

Roots: ___________ Roots: ___________ Roots: ___________ real imaginary real imaginary real imaginary

Find the value of the discriminant. Describe the number and type of roots:

7) 24 5y x x= − + 8) 2

4 5y x x= − − 9) 22 12y x x= − +

13

10) A quadratic equation has real rational roots. Which of the following could be the value of its

discriminant?

a. -10 b. 10 c. 25 d. 37

12) Factor: 38 27m − 13) Solve by factoring: 2

16 25 0x − =

Tell what intervals the following quadratics are increasing or decreasing.

14) 572

+−−= xxy inc: _________________ dec: _________________

15) 41132

−−= xxy inc: _________________ dec: _________________

16) What is the “b” value for the quadratic 1232

−+= bxxy if the roots are x=1 and x=-4?

14

Pre-AP Algebra 2 Name: ___________________________

4.8 Write Quadratic Equations

Ex 1) Write an equation for the parabola with the following roots in standard form. Graph your answer

to verify.

a) x = 3 and x = -8 b) x = 2 only c) x = 3i and x = -3i

d) x = 4+i and x = 4 – i e) x = 3+2i and x = 3 – 2i f) 1 2

,2 3

x x= = −

15

INCREASING: From left to right the graph is going up.

DECREASING: From left to right the graph is going down.

Ex 2) Write the intervals where f(x) is increasing and decreasing.

g) 192+−= xxy inc: ____________________ dec: ___________________

h) 472 2−+−= xxy inc: ____________________ dec: ___________________

Ex 3) Find the value of “b” in the equation 124 2−+= bxxy if the function has roots of

21

=x and 6−=x .

Ex 4) Over what interval(s) is the parabola in Ex 3 positive?

16

PAP Algebra II

4.8P Quadratics Practice Name: ______________________________

1. Graph the function.

y = -3(x +1)2 +4

Domain:_____________

Range:______________

Axis of Symmetry:______________

2. Given the factored form of the equation: y = -4(x -5) (x +3)

a) identify the root/s

b) convert to standard form

c) identify the y-intercept

3. Consider the graphs of the following equations. List the equations in order from

narrowest graph to widest graph.

a) y = -4x2 b) y =

1

3x

2 c) y =

5

3x

2 d) y = -x

2

b=_____ 4. Find the value of “b” in the equation 32

−+= bxxy if the vertex is (3, -12).

5. Write the equation for this parabola in

a) vertex form:

b) factored form:

c) standard form:

17

6. Write equation for the path of the golf ball in:

a) vertex form:

b) factored form:

c) standard form:

7. A rocket is shot straight up into the air from the top of a 30 foot cliff with an initial

velocity of 23 / secft . The equation for the path of the rocket is 2

16 23 30h t t= − + + .

a) What was the maximum height of the rocket?

b) When did the rocket reach it’s maximum height?

c) When did the rocket land on the ground?

d) How long after it was shot did it land on the ground?

e) What is a realistic domain for this situation?

8. Graph the following quadratic inequalities. Be sure to label at least 3 points.

a) 2

3( 5) 4y x< − − + b) ( )2

1

42 3y x≥ −

18

9. Find the vertex of the following parabolas.

V=_______ a) 962

−+−= xxy V=_______ b) 52432

−+−= xxy

10. A football is kicked from the ground at the 10 yard line, and it lands on the 50 yard

line. If the maximum height of the football is 80 yards, what is the equation for the

path of the football?

F(x)=______________________

V(x)=______________________

S(x)=______________________

11. Find all roots of the following parabolas. Simplify completely! Put all complex roots in

the form a+bi.

a) 232

−+−= xxy b) 3242

−+−= xxy

Roots: ___________ Roots: ___________

12. Write the equation in standard form for the parabola with roots:

__________________ a) x= -9 only __________________ b) 7x i= ±

_________________ c) 2 5x i= ±

19

13. Find the best fit quadratic equation to fit through the following data points.

(round to 3 decimal places)

a) Best fit quadratic: _____________________________

b) What is the maximum height of the object?

c) When does the object hit the ground?

14. A golf ball is hit from the top of a hill that is 16 feet high. It reaches it’s maximum

height of 87 feet 9 seconds after it was hit.

a) What is the equation for the path of the golf ball? (round to 3 decimals)

V(x)=___________________________

S(x)=___________________________

b) When does the golf ball hit the ground?

15. Use interval notation to identify what intervals the parabola 5432

+−= xxy is

increasing and decreasing.

Increasing: _________________

Decreasing: _________________

16. Factor the following quadratics completely and find the roots:

a) 216y x= − factored: ________________ roots: _________

b) 210 24y x x= − + factored: ________________ roots: _________

c) 23 13 10y x x= + − factored: ________________ roots: _________

TIME(s) 0 1 2 3 4 5 6

HEIGHT(m) 0.7 6.7 8.9 9.5 9.1 8.7 5.9

20

d) 2

4 19 5y x x= − + + factored: ________________ roots: _________

e) 2

3 10 8y x x= − + − factored: ________________ roots: _________

f) 2

5 23 10y x x= − − factored: ________________ roots: _________

17. Write the equation and graph the area of the set of all right triangles whose base is

three more units than the height.

Area = _______________

What is a reasonable domain and range

for this situation?

Domain: ____________

Range: _____________

18. Determine whether each equation has no solution, one real solution, two real solutions

or two complex solutions.

I. 20 5 1x x= + +

a) no solution b) one real solution c) two real solutions d) two complex solutions

II. 20 6 9x x= + +


III. 20 3 5x x= + +


19. Fill in the table for g(x) given that ( ) 2 ( 3) 3g x f x= − + .

x 1 2 3 4 5 6

g(x)

x -2 -1 0 1 2 3

f(x) 5 -3 7 6 -4 0

21

Pre-AP Algebra 2

WS 4.8 Writing Quadratics Equations Name: _____________________________

1. Write the equation of a quadratic with solutions at -3 and 15.

2. Write the equation of a quadratic with an a value of 3 and solutions at -3 and 4

3.

3. Write a quadratic equation of the form 2 0ax bx c+ + = that has roots 8 and 11.

4. Write the equation of a quadratic given the following graph.

5. A rectangular enclosure at a zoo is 35 feet long by 18 feet wide. The zoo wants to double

the area of the enclosure by adding the same distance, x, to the length and the width. Write

and solve an equation to find the value of x. What are the new dimensions of the enclosure?

6. Write the equation of the quadratic with solutions at 3 6i± .

7. Write the equation of the quadratic with solutions at 4 4i± .

8. Write the equation of a quadratic whose discriminant is -16 with an a value of 1 and a b value

of -18.

22

9. Write the equation of the quadratic in the graph shown.

10. Find two consecutive odd integers whose product is 79 greater than their sum.

11. The length of a rectangular picture frame is 6 inches less than twice the width. Find the width of the

frame if the area of the picture frame is 80 square inches.

12. Suppose you transform a square corn field by increasing one side by 2 units and decreasing the other side

by 3 units. If the area of the resulting corn field equals 24, what was the length of the original corn field?

13. A rectangle is 5 inches wide and 12 inches long. How much should be added to the width of the rectangle

to increase the diagonal by 7 inches?

14. Find three consecutive even integers such that the product of the first and third exceeds the product of the

second and 8 by 16.

23

Pre-AP Algebra 2 Name: ______________________________

Notes 4.9 Solve and Graph Quadratic Inequalities

1. a. Given ( )2

1 4y x≤ + − , what are the transformations being applied?

b. Would you use a solid or dotted line to graph?

c. Sketch the parabola on the graph to the right.

b. Using the inequality 2( 1) 4y x≤ + − , test the following ordered

pairs to determine where shading of the solution set should occur.

(−3,0) (−1,−1) (3, −2)

c. What ordered pair is usually an excellent test point? Why? When would this point not work to

check for the solution region?

2. a. Given 22 2 12y x x≥ + − , where are the roots? The vertex?

b. Would you use a solid or dotted line to graph? Why?

c. Sketch the parabola on the graph to the right.

d. Using the inequality 22 2 12y x x≥ + − , test the following ordered

pairs to determine where shading of the solution set should occur.

(0, −15) (0,0) (2,0)

Sketching the Graph of a Quadratic Inequality:

1) Sketch the graph of the parabola y = ax2 + bx + c. Use a solid line for ≥≥≥≥ ≤≤≤≤ and a dashed line for >>>> <<<<.

2) Test one point inside the U-shaped and one outside.

3) Only one of the test points will be the solution. Shade that region.

Ex. 1: Sketch the solution for y ≤ 3x2 – 6x

25

Ex. 2: Sketch the solution for y ≤ -x2 + 2x + 4

Word Problems:

1. The path of a football kicked from the ground can be modeled by xxh 2.102.2

+−= where ‘h’ is the

height (in yards) and ‘x’ is the horizontal distance (in yards) from where the ball is kicked. The

crossbar on a field goal post is 10 feet above the ground.

a. Write an inequality to find the values of x where the ball is high enough to go over the

crossbar.

b. Solve the inequality.

c. A player attempts to kick a field goal from 52 yards away. Will the ball have enough height

to go over the crossbar from this distance?

2. The arch of the Sydney Harbor Bridge in Sydney, Australia, can be modeled by

xxy 06.100211. 2+−= where ‘x’ is the distance in meters from the left pylons and ‘y’ is the height

in meter of the arch above the water. If the road is 52 meters above the water, for what distances x

is the arch above the road?

26

x

y

x

y

Pre-AP Algebra 2 Name: ___________________________

WS 4.9 Solve and Graph Quadratic Inequalities

Using your graphing calculator, plot at least three accurate points to form the boundary line.

Shade the correct region and then list 2 ordered pairs that could be solutions.

1. 3)2( 2+−≥ xy 2 6)1(3 2

++−> xy

3 .Which of the points given would satisfy the inequality? There can be more than one answer.

1222 2−+≥ xxy

a. (1, 3) b. (2, 0) c. (5, -2) d. (-2, 4)

Application Problems

4) You decide to start at the left base of a mountain and show off your hiking skills. The mountain might

be represented by the function y = -.0026(x – 310)2 + 250. You need to use oxygen above 200 feet. You

are asked to analyze the part of the path that is safe to hike without oxygen?

a. Write the inequality

b. Solve for reasonable solutions.

c. Identify the vertex of the mountain

d. Describe the window used on your graphing calculator.

5) The Arc-de-Triomphe in Paris might use the equation y = -.444x2 + 13.32x + .1, to make the arch at the

top. A row of decorative stone is 85 feet above the ground. Painters need you to analyze the area above

the stone so they can paint that area.

a. Write the inequality _________________

b. Solve for reasonable solutions.______________

c. Identify the vertex of the arch_________________

X min=

X max=

Y min=

Y max=

27

6) The Eiffel tower in Paris has four arches at the base. Each one has an equation similar to

y = -.02x2 + 3x – 2.5. If they are starting at the right and left bases and painting up the arch, what domain

will get painted if the need to stay under 65 feet?


b. Write the domain of the arch that gets painted on both sides?

7) The Grand Canyon has valleys shaped like parabolas. The inequality 4008.10006. 2−−≥ xxy

represents the part of a canyon that can be seen from an observation tower.

a. Find a good fit window._______________________________

b. Write the domain of the valley that is below sea level?_______________

(allowing negative values)

c. Explain how you decided on a good fit window.

8) A plane is flying over the Grand Canyon. It follows the function f(x) = -2.5(x – 1000)2 + 25000, where x

is minutes and f(x) is height in feet. For what times, x, is the plane above 20,000 feet?


b. Write the times of the plane?_______________

9) A picture is 12 inches by 14 inches. You are going to add a frame that is x inches thick. You want the

resulting framed picture to have an area less than or equal to 500 square inches. What is a reasonable

solution for the thickness of the frame? (Assume the frame must be at least 1 inch

thick)

a. Write the inequality______________________

b. What is a reasonable solution?___________________

10) Twenty-eight less than the square of some number is 3 times the same number. Find the number. Is

there more than one possibility?

28

Name: ____________________________________ Date: __________________

Algebra 1, Unit #6 – Quadratic Algebra – L16 The Arlington Algebra Project, LaGrangeville, NY 12540

2 5

2 15

y x

x y

= ++ =

Solving Linear – Quadratic Systems Algebraically Algebra 1

In this lesson we will begin to work with solving linear-quadratic systems of equations. Recall that to solve a system we must find the set of all points ( ),x y that satisfy all equations in the system. We

will review this concept with an example from linear systems. Exercise #1: Consider the linear system shown to the right.

(a) Solve this system algebraically using the (b) Explain, in graphical terms, what the ordered substitution method. pair from (a) represents.

The substitution method was used above because it is the only method that we can use to solve linear – quadratic systems algebraically. Solving such systems requires solving a quadratic equation. Since we are working with quadratics, it is natural to expect more than one answer. This has a graphical connection as Exercise #2 will illustrate. Exercise #2: Consider the sketch of a line and a parabola shown at the right. (a) What is the maximum number of intersection points that a line and a

parabola could have? Illustrate with a picture. (b) What is the minimum number of intersection points that a line and a

parabola could have? Illustrate with a picture. (c) Is it possible for a line and a parabola to intersect in only one point? If

so, illustrate with a picture.

29


Exercise #3: Solve each of the following systems of equations algebraically and check using STORE on your calculator. In each case the substitution method should be used to begin the process.

(a) 2 4 17 9

y x xy x

= + −= +

(b) 2 2 76 3

y x xy x

= + += +

(c) 2 2 63 12y x xx y= + −+ = −

(d) 2

10 5

7 5

y x

y x x

− == + +

30

Name: ____________________________________ Date: __________________


Solving Linear – Quadratic Systems Algebraically Algebra 1 Homework

Skills 1. Which of the following is a solution to the system of equations shown to the right? (1) ( )4, 7 (3) ( )3, 0

(2) ( )4, 1− − (4) ( )2, 5

2. Mateo produced the following table on his calculator to find the solutions to a linear-quadratic

system of equations. Based on this table, which of the following sets gives the x-values that solve this system?

(1) { 4, 2}− (3) {3, 6} (2) { 4, 3}− (4) { 2,1}− 3. Which of the following is not a possible number of solutions to a linear-quadratic system? (1) 1 (3) 3 (2) 2 (4) 0 Solve each of the following linear – quadratic systems of equations algebraically and check using STORE on your calculator. 4. 2 5 2

2y x xy x

= + −= −

5. 2 3 33 6

y x xy x

= − +− = −

2 9

3

y x

y x

= −= +

31


6. 2 2 84 5y x xx y= + −− =

7. 2

2 10

2 2

x y

y x x

− = −= − −

Applications 8. The price C, in dollars per share, of a high-tech stock has fluctuated over a twelve-year period

according to the equation 214 12 ,C x x= + − where x is in years. The price C, in dollars per share, of a second high-tech stock has shown a steady increase during the same time period according to the relationship 2 30.C x= +

(a) For what values are the two stock prices the same? (Only an algebraic solution will be accepted.) (b) Determine the values of x for which the quadratic stock price is greater than the linear stock price.

State your answer as an inequality. (Hint: You should be able to answer this almost immediately based upon your analysis in part (a) above.)

Reasoning

9. Which value below for b would result in the linear-quadratic system 2 3 1y x x= + + and y x b= − − having only one intersection point? Justify your answer algebraically, graphically or with a table.

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

32

pap algebra 2 unit 4b - hhs algebra ii -...

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