Download Cubes and Cube Roots

Quadratics
Objectives Algebra
1. The student will express the square roots and cube roots of whole numbers and the square root
of a monomial algebraic expression in simplest radical form.
2. To solve quadratic equations algebraically and graphically.
3. To solve analyze function (linear and quadratic) families and their characteristics both
algebraically and graphically, including a) determining whether a relation is a function; b) domain
and range; c) zeroes of a function; d)x- and y-intercepts; e)finding the values of a function for
elements in its domain; and f)making connections between and among multiple representations of
functions including concrete, verbal, numeric, graphic, and algebraic problems that can be
represented by quadratic equations
Square Roots & Cube Roots
1. Squares and square roots are opposites.
Cubes and cube roots are opposites.
72 = 49 and √ 49 = 7
2. Radical
√ 27
3. √15
√5●3●3
_
3
√ 2● 2 ● 2 ● 3
4.
3
√24
23 = 8 and
Radical Sign
√
3
√8 =2
Radicand 27
3 √5
23√3
Solving Quadratic Equations
The solution to a quadratic equation is called the ”answer”, the “solution”, the “roots”,
the “real roots”, the” zeros” , or the “x-intercepts” of the equation.
You may have “no” solutions, “1” solution, or “2” solutions.
1. Standard Form of a Quadratic Equation
ax2 + bx + c = 0
3x2 + 2x - 5 = 0
Quadratic Term 3x2
a= 3
Linear Term
+ 2x
b= 2
Constant Term
-5
c=-5
2. Zero Product Property
If ab = 0, then a = 0 or b = 0.
a) To solve quadratic equations, set each factor equal to zero.
b) Then solve each equation.
c) The value of the variable is the number that will make the factor become a zero.
3. Memorize the Quadratic Formula.
x = - b ± √ b2 - 4ac
2a
a) Write the equations in standard form: ax2 + bx + c = 0.
If necessary, multiply both sides by the LCD to clear fractions.
b) Write down the values for a, b and c.
c) Substitute the values for a, b and c into the quadratic formula.
d) Simplify
e) Check the answers.
4. Graphing - y = ax2 + bx + c
The graph of a quadratic equation is a parabola.  
The zeros of a function are the values of x when y = 0.
Let x = 0 and solve for y, These are the points where the graph crosses the x-axis.
If the line crosses the x-axis in two places, there are 2 answers.
If the line crosses the x-axis in one place, there is one answer.
If the line does not cross the x-axis, there are no answers.
Which way does curve go?
y = ax2 + bx + c
If a is positive, the graph goes up. On this graph, the lowest point is the vertex. 
If a is negative, the graph goes down. On this graph, the highest point is the vertex. 
5. The axis of symmetry is the vertical line through the vertex.
x= -b
2a
To plot the vertex, plug in the values of a and b and solve for x.
Now plug in the value of x (into the quadratic equation) and solve for y.
(x,y) are the coordinates of the highest or lowest point of the parabola.
6. The discriminant,
b2 - 4ac, is the radicand of the quadratic formula.
The number of solutions can be found using the discriminant.
Discriminant
1) b2 - 4ac > 0 (positive)
2
Solution
two real roots
2) b - 4ac = 0 (zero)
one real root
3) b2 - 4ac < 0 (negative)
no real roots
7. The standard form of a quadratic equation
y = a(x - h)2 + k
(h, k) are the coordinates of the vertex of the parabola.
The value of a makes the parabola open or close.
If a is positive, the graph curves upward.
If a is negative, the graph curves downward.
The value of h makes the parabola move to the right or left.
The value of k makes the parabola move up or down.
Who’s Perfect?
12 = 1
√1 = 1
2,3
22 = 4
√4 = 2
5,6,7,8
32 = 9
√9 = 3
10,11,12,13,14,15
42 = 16
√16 = 4
17,18,19,20,21,22,23,24
52 = 25
√25 = 5
26,27,28,29,30,31,32,33,34,35
62 = 36
√36 = 6
37,38,39,40,41,42,43,44,45,46,47,48
72 = 49
√49 = 7
50,51,52,53,54,55,56,57,58,59,60,61,62,63
82 = 64
√64 = 8
65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80
92 = 81
√81 = 9
82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99
102 = 100
√100 = 10
101, 102, 103, 104, 105, 106, 107, 108, 109, 110,
111, 112, 113, 114, 115, 116, 117, 118, 119, 120
112 = 121
√121 = 11
122 = 144
√144 = 12
132 = 169
√169 = 13
142 = 196
√196 = 14
152 = 225
√225 = 15
Square Roots
Squaring a number means using that number as a
factor two times.
82 = 8 • 8 = 64
(-8)2 = -8 • –8 = 64
Remember to use ( ) because
-82 = -8 • 8 = -64 not 64
The opposite of squaring a number is finding the
square root.
To find the square root of 64, you must find
two equal factors whose product is 64.
x2 = x • x = 64
Since 8•8 = 64, one square root of 64 is 8.
What is another square root of 64?
√64 is called a radical expression
√ is called the radical sign. It indicates the
nonnegative or principal square root of the
expression.
64 is called the radicand
√64 = 8 indicates the principal square root.
-√64 = -8 indicates the negative square root.
± √64 = ±8 indicates both square roots of 64.
√-64 is not possible yet.
√49 = √7 • 7 = 7
√100 = √10 • 10 = 10
√10 = √2 • 5 • 2 • 5 = 10
√12 = √2 • 2 • 3 = 2√3
3√32 = 3√2•2•2•2•2 = 3•2•2 √2 = 12√2
√8x3y4 =
√2 •2 •2 xxx yyyy
2xy2√2x
2√9x6y10 =
2√3•3 xxxxxx yyyyyyyyyy
2•3x3y5
6x3y5
√121x8 =
√11•11 x4 •x4
11 x4
√-25
Simplify radicals by taking out all
perfect squares.
√8 = √2 •2 •2 = 2√2
√8 = √2 •4 = 2√2
-√12 = -√2 •2 •3 = -2√3
-√12 = -√4 •3 = -2√3
±√27x5 = ±√3 •3 •3 •x2 •x2 •x = ±3x2√3x
±√27x5 = ±√3 •9 •x4 •x = ±3x2√3x
√50x5 = √5•5•2•x2•x2•x = 5x2√2x
√50x5 = √25•2•x4•x = 5x2√2x
Cubes and Cube Roots
To understand cube roots, first you must understand cubes ...
How to Cube A Number
To cube a number, just use it in a multiplication 3 times ...
Example: What is 3 Cubed?
3 Cubed =
= 3 × 3 × 3 = 27
Note: we write down "3 Cubed" as 33
(the little "3" means the number appears three times in multiplying)
Some More Cubes
4 cubed = 43 = 4 × 4 × 4 = 64
5 cubed = 53 = 5 × 5 × 5 = 125
6 cubed = 63 = 6 × 6 × 6 = 216
Cube Root
A cube root goes the other direction:
3 cubed is 27, so the cube root of 27 is 3
3
27
The cube root of a number is ...
... the special value that when cubed gives the original number.
The cube root of 27 is ...
... 3, because when 3 is cubed you get 27.
Note: When you see "root" think
"I know the tree, but what is the root that produced it?"
In this case the tree is "27", and the cube root is "3".
Here are some more cubes and cube roots:
4
64
5
125
6
216
Example: What is the Cube root of 125?
Well, we just happen to know that 125 = 5 × 5 × 5 (if you use 5 three times in a
multiplication you will get 125) ...
... so the answer is 5
The Cube Root Symbol
This is the special symbol that means "cube root", it is the
"radical" symbol (used for square roots) with a little three to
mean cube root.
You can use it like this:
(you would say "the cube root of 27 equals 3")
You Can Also Cube Negative Numbers
Have a look at this:
If you cube 5 you get 125:
If you cube -5 you get -125:
5 × 5 × 5 = 125
-5 × -5 × -5 = -125
So the cube root of -125 is -5
Perfect Cubes
The Perfect Cubes are the cubes of the whole numbers:
Perfect
Cubes:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
etc
1
8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 ...
It is easy to work out the cube root of a perfect cube, but it is really hard to work out other
cube roots.
Example: what is the cube root of 30?
Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4.

Let's try 3.5: 3.5 × 3.5 × 3.5 = 42.875

Let's try 3.2: 3.2 × 3.2 × 3.2 = 32.768

Let's try 3.1: 3.1 × 3.1 × 3.1 = 29.791
We are getting closer, but very slowly ... at this point, I get out my calculator and it says:
3.1072325059538588668776624275224
... but the digits just go on and on, without any pattern. So even the calculator's answer is
only an approximation !
Product Property
√a•√b = √ab
√6•√15 =
√2•3•3•5 =
3√2•5 = 3√10
3√12•1√5=
3√3•2•2•5= 6√15
√5•2√15 =
2√5•15 =
2√5•5•3 = 10√3
√10a•√20a2 =
√10•10•2•a•a2 or √2•5•2•5•2•a•a•a
2•5 a √2a
10a√2a
3√8•2√18= 3•2 √8•18=
6√2•4•2•9=
6•2•3•2=
72
15 + 20 =
5
15 + 20 =
5
5
3 + 4 = 7
15 + 20√3 =
5
15 + 20√3 =
5
5
3 + 4√3 =
10 + 5√3 =
5
10 + 5√3 =
5
5
2 + 1√3 =
10 + √12 =
2
10 + 2√3 =
2
2
5 + 1√3 =
14 + √18 =
2
14 + 3√2 =
2
2
14 + 3√2 =
2
12 + 3√20 =
2
12 + 6√5 =
2
2
6 + 3√5 =
Standard Form of a Quadratic Equation
ax2 + bx + c = 0
3x2 + 2x - 5 = 0
Quadratic Term is 3x2
Linear Term is + 2x
Constant Term is - 5
Ways to Solve a Quadratic Equation:
1. Factoring – if the constant term is 0
or the factors can be easily determined.
2. Solving equation – if the equation can
be simplified to x2 = a number.
3. Graphing the equation – use when an
approximate solution is sufficient.
4. Quadratic equation – other methods may
be easier but this method always works.
ZERO PRODUCT PROPERTY
If ab = 0, then a = 0 or b = 0.
To solve quadratic equations, set each
factor equal to zero.
a2 + 4a - 12 = 0
(a - 2)(a + 6) = 0
Then solve each equation.
(a - 2) = 0
a=2
| (a + 6) = 0
|
a=-6
The value of the variable is the number that will make
the factor become a zero. This is the solution.
a = 2 and a = - 6
Substitute these values into the factors. What happens?
(a - 2)(a + 6) = 0
Substitute these values into the original equation.
What happens?
a2 + 4a - 12 = 0
Using The Zero Product Property
1) x2 + 6x + 5 = 0
(x + 5)(x + 1) = 0
(x + 5) = 0 | (x + 1) = 0
x = -5 and x = -1
There are two answers.
2) x2 – 6x + 9 = 0
(x - 3)(x - 3) = 0
(x - 3) = 0 | (x - 3) = 0
x = 3 and x = 3
3)
There is one answer.
x2 – 2x + 3 = 0
There is no answer.
Using Square Roots
The standard form of a quadratic equation
is ax2 + bx + c = 0 where:
a is the “leading coefficient”
b is the coefficient of the linear term
c is the constant term
If b = 0, then ax2 + c = 0.
Since there is no linear term, isolate x2 and then
find the square root of both sides.
1) 3x2 - 48 = 0
3x2 = 48
x2 = 16
x=±4
2) 5x2 - 500 = 0
5x2 = 500
x2 = 100
x = ± 10
3) (x – 5)2 = 36
√(x – 5)(x – 5) = √36
x – 5 = ±6
x – 5 = 6 | x – 5 = -6
x = 11 & x = -1
4)
3(x + 1)2 = 12
(x + 1)2 = 4
√(x + 1)(x + 1) = √4
x + 1 = ±2
x+1=2|x+1=-2
x = 1 & x = -3
5) x2 = 81
6) x2 - 121 = 0
7) 25x2 = 9
8) 20x2 - 125 = 0
9) (x + 2)2 = 49
10) 2(x – 3)2 = 18
Not all quadratics equations of the form
ax2 + c = 0 can be solved algebraically
or graphically.
11) 4x2 + 12 = 0
12) 5x2 + 30 = 0
How Are The Three Graphs Related?
1)
y = x2 – x + 1
y = x2 – x + 3
y = x2 – x - 2
How does a change in the value of c change the
graph of y = ax2 + bx + c?
2) y = x2 + x + 1
y = ½x2 + x + 1
y = 2x2 + x + 1
How does a change in the value of a change the
graph of y = ax2 + bx + c?
3) y = x2 – 2x + 1
y = x2 – 4x + 4
y = x2 – 6x + 9
Quadratic Equation
1.
Zero Product Property
x2 + 6x + 5 = 0
(x + 5)(x + 1) = 0
(x + 5) = 0 | (x + 1) = 0
x = -5 and x = -1
Let y1 = x + 5 and y2 = x + 1.
Let y3 = x2 + 6x + 5
What is happening?
What is the solution?
{-5, -1}
2. x2 – 8x + 12 = 0
(x – 2)(x – 6) = 0
(x – 2) = 0 | (x – 6) = 0
x = 2 and x = 6
Let y1 = x - 2 and y2 = x - 6
Let y3 = x2 – 8x + 12
What is happening?
What is the solution?
{2, 6}
3. x2 + 10x + 21 = 0
(x + 7)(x + 3) = 0
(x + 7) = 0 | (x + 3) = 0
x = -7 and x = -3
What is the solution?
{-7, -3}
4. x2 – 3x –10 = 0
(x + 2)(x - 5) = 0
(x + 2) = 0 | (x - 5) = 0
x = -2 and x = 5
What is the solution?
5.
{-2, 5}
x2 – 16 = 0
(x - 4)(x + 4) = 0
(x - 4) = 0 | (x + 4) = 0
x = 4 and x = - 4
What is the solution?
{4, - 4}
Trinomial Form
1. x2 + 6x + 5 = 0
2. x2 – 8x + 12 = 0
Factors
(x + 5)(x + 1) = 0
(x – 2)(x – 6) = 0
3. x2 + 10x + 21 = 0 (x + 7)(x + 3)= 0
4. x2 – 3x –10 = 0
(x + 2)(x – 5) = 0
Solutions
{-5, -1}
{2, 6}
{-7, -3}
{-2, 5}
5. x2 – 16 = 0
(x – 4)(x + 4) = 0
{4, - 4}
6.
(x + 4)(x + 2) = 0
{- 4, - 2}
x2 + 6x + 8 = 0
7. 2x2 + 12x + 16 = 0 2(x + 4)(x + 2) = 0 {- 4, - 2}
8. 5x2 + 30x + 40 = 0 5(x + 4)(x + 2) = 0 {- 4, - 2}
9. 2x2 + 7x + 3 = 0
(2x + 1)(x + 3) = 0 {- 1/2, - 3}
10 3x2 + 14x + 8 = 0
(3x + 2)(x + 4) = 0 {- 2/3, - 4}
1. The solutions of a quadratic equation
are the _____________ of the
graph of the equation.
2. The solution to a quadratic equation
is also called _____________
3. Write a quadratic equation with
a solution set { 4, -7 }.
GRAPHING A QUADRATIC
The graph of a quadratic equation is a parabola.
y = ax2 + bx + c
If a is positive, the graph goes up. On this
graph, the lowest point is the vertex.
If a is negative, the graph goes down. On this
graph, the highest point is the vertex.
Plot the vertex. x = - b and plug in the value of x
to solve for y.
2a
The axis of symmetry is the vertical line x = - b
2a
Make a table of values using x-values to the left
and right of the axis of symmetry.
GRAPHING A QUADRATIC
Graph
y = 2x2 + 4x - 3
a=
b=
c=
a is positive so the graph goes up.
Find the coordinates of the vertex (x, y).
Use x = - b to find the x-coordinate.
2a
x=-4
x = -1
2(2)
Substitute the value of x into the original equation
and solve for y.
Y = 2x2 + 4x - 3
Y = 2(-1)2 + 4(-1) - 3
Y=-5
The vertex is ( -1, -5 )
Note: The axis of symmetry is x = -1
Solving Quadratic Equations
Identify the values of a, b, and c.
1. 2x2 - 7x + 3 = 0
a=
b=
c=
2. 3x2 + 2x – 5 = 0
a=
b=
c=
3. 6x2 + 2x = 0
a=
b=
c=
4. x2 + 3x – 9 = 0
a=
b=
c=
5. 8x2 - x + 3 = 0
a=
b=
c=
6. x2 – 25 = 0
a=
b=
c=
Write the equation in standard form.Then
identify the values of a, b, and c.
1. 4x2 = 48
a=
b=
c=
2. 2x2 - 3x = 8
a=
b=
c=
3. x2 = - 5x + 14
a=
b=
c=
4. - 2x + 5 = x2
a=
b=
c=
5. 3x2 = - 15x
a=
b=
c=
6.
2x2 + 1x = - 3
3
4
a=
b=
c=
QUADRATIC FORMULA
1. The quadratic formula
x = - b ± √b2 - 4ac
2a
2. To use the quadratic formula to solve
quadratic equations:
a) Write the equation in standard form:
ax2 + bx + c = 0.
b) To clear fractions multiply both sides by the LCD.
c) Write down the values for a, b and c.
a=
b=
c=
d) Substitute the values of a, b & c into
the quadratic formula.
e) Simplify.
f) Check your answers.
DISCRIMINANTS
1. The discriminant is the radicand of the
quadratic formula. b2 − 4ac
2. Use the discriminant to find the type and
number of solutions.
Discriminant
Solution
b2 − 4ac is positive
two solutions
b2 − 4ac is zero
one solution
b2 − 4ac is negative
no solution
Find the discriminant & number of solutions.
1) x2 – 2x + 3 = 0
2) x2 – 2x + 1 = 0
3) x2 – 2x - 2 = 0
GRAPHING QUADRATICS
1. The graph of a quadratic equation is a parabola.
2. The standard form of a quadratic equation:
y = a(x - h)2 + k
(h,k) are the coordinates of the vertex
3. If “a” is positive, the graph curves upward.
If “a” is negative, the graph curves downward.
4. If “a” is > 0, the curve closes.
If 0 < a < 1, the curve opens.
5. If “k” is > 0, then the graph moves upward.
If “k” is < 0, then the graph moves downward.
6. If “h” is > 0, it moves the graph to the right.
If “h” is < 0, it moves the graph to the left.
BALLET PARABOLAS
y = a(x - h)2 + k
The value of “a” opens or closes the
a
parabola.
if a > 0 , the curve closes
if 0 < a < 1, the curve opens
The value of “h” moves the parabola right or left
h
if h > 0, it moves to the right
if h < 0, it moves to the left
The value of “k” moves the parabola up or down.
k
if k > 0, it moves up
if k < 0, it moves down
Which way does the parabola curve?
If “a” is positive, the graph curves upward.
If “a” is negative, the graph curves downward.
1) y = x2
2) y = 3x
2
3) y = .5x2
4) y = x - 3
2
5) y = x2 + 3
6) y = 3x - 2
2
7) y = .5x2 + 3
8) y = 6x + 2
2
9) y = .8x2 - 1
10) y = .2x - 2
2
11) y = 3x2 - 4
12) y = (x – 2)
2
13) y = (x + 3)2
14) y = .5(x + 1)
2
15) y = 3(x – 2)2
16) y = (x – 3) + 2
2
17) y = (x + 3)2 - 2
18) y = 5(x – 2) + 3
2
19) y = .2(x + 3)2 - 2
20) y = .5(x – 2) + 1
2
21) y = - 3x2 + 1
22) y = - 5x - 2
2
23) y = 6x2 + 2
24) y = .5(x – 1) - 2
2
25) y = -2(x + 3)2 - 3
26) y = 4(x – 2) - 1
2
27) y = - 3(x – 1)2 - 3
28) y = .2(x – 4) + 1
2
29) y = - 3(x + 2)2 - 1
30) y = - .5(x + 3) + 2
2
WORD PROBLEMS - QUADRATICS
1. The height h in feet of a ball t seconds after being tossed upward is given by the
Formula h = 96t - 16t2
A. After how many seconds will the ball reach a height of 80 ft for the second time?
B. After how many seconds will it hit the ground?
C. What is its maximum height?
2. Heather has 120 meters of fence to make a rectangular pen for her rabbits. If a shed is used
as one side of the pen, what would be the maximum area for the pen?
3. A flare is launched from a life raft with an initial upward velocity of 144 ft per sec. How many
seconds will it take for the flare to return to the sea? Use h = vt - 16t2
4. The length of Rachel's rectangular garden is 5 yards more than its width. The area of the
garden is 234 square yards. What are its dimensions?
5. A missile is fired with an initial upward velocity of 2,320 feet per second. When will it reach
an altitude of 40,000 feet? Use the formula h = vt - 16t2
6. A flare is launched with an initial upward velocity of 128 feet per second. How long will it take
for the flare to hit the ground? Use the formula: h = vt - 16t2
7. The width of a rectangle is 55 cm less than three times its length. The area of the rectangle
is 100 cm2. Find the dimensions of the rectangle.
8. A certain fireworks rocket is set off at an initial upward velocity of 440 ft per sec. These
fireworks are designed to explode at a height of 3,000 feet. How many seconds after it is set off
will the rocket reach 3000 feet and explode. Use h = vt - 16t2
Names _________________________and_______________________________
THE POPCORN BOX PROBLEM
Objective: To construct an open box (no lid) that will have the largest possible volume.
Materials: Grid paper (18 X 24), tape, scissors and a graphing calculator for each group.
Procedure: Each group will cut out congruent squares from the four corners of the 18 X 24 rectangle.
The sides are then turned up and taped to make a box with no lid. Complete the appropriate part of
the table using your box. Then turn in the box to your teacher.
1. You have been assigned a cut out square of _________cm.
2. Which box do you think has the largest volume?. The box with cut out square of 1X1 cm, 2X2 cm,
3X3 cm, etc. If you believe that have the same volume, state that they are the same. _________
3. Assemble your box. Turn in your completed box to your teacher. With all the boxes on display,
which box do you think has the greatest volume? _____________
4. Complete the following table.
Length of
square (x)
Area of corner cutout (cm2)
1 cm
12 =
2 cm
22 =
3 cm
32 =
4 cm
42 =
5 cm
52 =
6 cm
62 =
7 cm
72 =
8 cm
82 =
X cm
X2 =
Length
(cm)
Width
(cm)
Height
(cm)
Volume
(cm3)
5. Scatter Graph - Using the graphing calculator, go to Stat, list and let L1 be (the value of x) the length
of the square in the cut-out areas and let L2 be the volume of the box. Input data from table above and
graph a scatter plot. Use trace and record the greatest volume. ___________
6. Function - Set window.
x min: -2
x max: 16
x scl: 1
y min: -100
y max: 660
y scl: 20
Let x = length of cut-out square and y = volume. Place the formula for the volume in y1 = using data
in last row of table above. Graph and use trace to find the greatest volume. Go to table and find tine
greatest volume. Change table set to .1 (tenths) and record the box with the greatest volume. ______
7. Evaluate expression - Go to home screen. Store values for x and find the volume using formula
from last row of table. What is the greatest volume? ________