unit 4 notes
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Unit 4 Notebook 2012.notebook
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Nov 59:17 AM
Irrational Expressions
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Simplifying Radicals
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Did you know...
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Do you know your perfect squares???List the first 15 perfect squares...
1 814 1009 12116 14425 16936 19649 22564
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Can you simplify the following rational expressions? Explain why each is rational!
149162536496481100121144169196225...
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What if the expressions were IRRATIONAL? Do you know how to simplify these?
149162536496481100121144169196225...
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Steps to Simplifying Radicals:
1. Rewrite the irrational expression as the product of 2 radicals. One of those radicals MUST have a perfect square inside it.
2. Break down the radical with the perfect square inside it.
3. Write your final answer.
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How would you simplify a radical that has a variable inside of it???
Can you simplify the following:
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Can you simplify the following irrational expressions?
149162536496481100121144169196225...
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Can you simplify the following irrational expressions?
149162536496481100121144169196225...
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Homework: p. 1‐2 #1‐ 21 odds
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#3 #7
#11 #15
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Answers to p. 1‐ 2 #1‐ 21 odds
1. 3.
5. 7.
9. 11.
13. 15.
17. 19.
21.
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Rationalizing Radicals
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So we know how to simplify ...
Do you know WHY we have to simplify
149162536496481100121144169196225...
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Rationalizing a denominator: Removing all radical symbols from the denominator of a fraction.
Does anyone know the best way to get rid of the radical symbol in our example?
149162536496481100121144169196225...
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Steps to RATIONALIZING a fraction:1. Simplify the expression if possible by dividing.2. Multiply the top and bottom by the radical in the denominator.3. Simplify the remaining fraction.
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Rationalizing each expression below:149162536496481100121144169196225...
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Rationalizing each expression below:149162536496481100121144169196225...
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Homework: p. 3 #1- 8
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#2 #4
#6 #8
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Answers to p. 3 #1- 8
1. 2.
3. 4.
5. 6.
7. 8.
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Adding & Subtracting Radicals
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Do you recall how to simplify the following:
1. 2x + 3x
2. -5xy - 8xy
3. 7x2y - 17x2y + 9xy2
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Then you should be able to simplify the following:
1.
2.
3.
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When you add or subtract radicals, you must have the same # inside the radical
symbol (radicand).
It's just like combining like terms!!!
Ex:
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Use a bit of logic to do these problems. Simplify the radical that is easier to YOU first! Then you
know what the "last name" has to be for the other radical.
Ex: Which radical is easier to break down?
What is the "last name" going to be?
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Simplify:1. 2.
3. 4.
5. 6.
149162536496481100121144169196225...
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Homework:p. 4‐ 5 # 7‐ 27 odds
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#9 #15
#17 #21
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Answers to p. 4 ‐5, #7‐ 27 odds
7. 9. 11.
13. 15. 17.
19. 21. 23.
25. 27.
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QUIZ #6 TODAYIf you can simplify the following problems, then you should be ok ;)
1. 2. 3. 4.
149162536496481100121144169196225...
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QUIZ #6 Do NOT do #2, 4 & 7
CR #6 due tomorrow
BONUS (3 points)Simplify:
149162536496481100121144169196225...
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Journal Entry- Friday, Nov. 30
Would you like to write a short paper on a mathematician and create a poster based on their "findings" to count as a test?
Would you like to work in groups (3 people max) or by yourself?
HAPPY FRIDAY!
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Multiplying Radicals
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What #'s are on the OUTSIDE?
What 3's are on the INSIDE?
Steps to Multiplying Radicals:
1. Multiply OUTSIDES with OUTSIDES2. Multiply INSIDES with INSIDES.3. Simplify if possible.
Example:
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Try these. Multiply and simplify:
1. 2.
3. 4.
149162536496481100121144169196225...
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Uh oh‐ these have variables! Multiply and simplify:
1. 2.
3. 4.
149162536496481100121144169196225...
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Using the Distributive Property:(Remember, OUTSIDES with OUTSIDES, INSIDES with INSIDES)
1. 2.
3. 4.
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Homework:p. 6 # 1‐ 14
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p. 6 #1 14#3 #6
#9 #12
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Answers to p. 6 # 1‐ 14
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14.
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FOIL-ing Radicals
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Do you recall how to multiply binomials? You guys call this the FOIL-ing Method.
Multiply and simplify: 1. (x + 2) (x - 9) =
2. (3 - x) (3 + x) =
3. (x2 - 1) (x2 - 7) =
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Can you apply the FOIL-ing Method to RADICALS??? Remember OUTSIDES with OUTSIDES and INSIDES with INSIDES!
Multiply and simplify: 1. =
2. =
149162536496481100121144169196225...
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Multiply and simplify: 1. 2.
3.
149162536496481100121144169196225...
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Homework:p. 7 #21 28
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Answers to p. 7 #21 28
21. 22.
23. 24.
25. 26.
27. 28.
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Activity?
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Pythagorean Theorem
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10 8 6 4 2 0 2 4 6 8 10
1098765432
12345678910
x
y
T
C Mrs. E can't find her car in the parking lot after a long trip at Target. Mrs. E is
standing at T(-4, 2). After much ado, she sees her car is parked at C (4, 8). Can you determine how far away from her car she
is???
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Mrs. E can't find her car in the parking lot after a long trip at Target. Mrs. E is
standing at T(-4, 2). After much ado, she sees her car is parked at C (4, 8). Can you determine how far away from her car she
is???
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DID YOU KNOW???The DISTANCE formula was "created" by
manipulating the Pythagorean Theorem.
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Pythagoras (569-500 B.C.E.) was born on the island of Samos in Greece, and did much traveling through Egypt, learning, among other things, mathematics. Not much more is known of his early years. Pythagoras gained his famous status
by founding a group, the Brotherhood of Pythagoreans, which was devoted to the study of mathematics. The group was almost cult-like in that it had symbols,
rituals and prayers. In addition, Pythagoras believed that "Number rules the universe,"and the Pythagoreans gave numerical values to many objects and ideas.
These numerical values, in turn, were endowed with mystical and spiritual qualities.
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Legend has it that upon completion of his famous theorem, Pythagoras sacrificed 100 oxen. Although he is credited with the discovery of the famous theorem, it is not possible to tell if Pythagoras is the actual author. The Pythagoreans wrote many geometric proofs, but it is difficult to ascertain
who proved what, as the group wanted to keep their findings secret. Unfortunately, this vow of secrecy prevented an important mathematical idea from being made public. The Pythagoreans had discovered irrational numbers!
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If we take an isosceles right triangle with legs of measure 1, the hypotenuse will measure sqrt 2. But this number cannot be expressed as a length that can be
measured with a ruler divided into fractional parts, and that deeply disturbed the Pythagoreans, who believed that "All is number." They called these numbers
"alogon," which means "unutterable." So shocked were the Pythagoreans by these numbers, they put to death a member who dared to mention their
existence to the public. It would be 200 years later that the Greek mathematician Eudoxus developed a way to deal with these unutterable
numbers...
1
1
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The Pythagorean Theorem is used to find the length of a side of a
RIGHT TRIANGLE.
a2 + b2 = c2
a & b represent the length of the legs
c represents the length of the hypotenuse
***The hypotenuse is the longest side of the right triangle.
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The Pythagorean Theorem also states:
the area of square A plus the area of square B is equal to the area of square C.
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Pythagorean Triples:Common lengths of sides in a right triangle.
I bet you know at least one of these triples....
( 3 , 4 , 5 )
( 5, 12, 13)
( 7, 24, 25)
( 8, 15, 17)
( 9, 40, 41)
(11, 60, 61)
(12, 35, 37)
(13, 84, 85)
(16, 63, 65)
(20, 21, 29)
(28, 45, 53)
(33, 56, 65)
(36, 77, 85)
(39, 80, 89)
(48, 55, 73)
(65, 72, 97)
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How do you tell if any given 3 sides would form a right triangle? Use the
Pythagorean Theorem to check!!!
Determine if the following lengths form a right triangle.
1. 2.10
24
26 7
14
15
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Find the missing side of the right triangle and round to the nearest tenth.
26
3 101
4
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Find the missing side of the right triangle and leave your answer in simplest radical form.
12
5
2 6
15
5
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Homework: p. 8 #1- 12
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Answers to p. 8 #1- 12
1. NO 2. YES 3. YES
4. YES 5. NO 6. YES
7. 8.9 8. 6.7
9. 12.2 10. 7.6
11. 7.3 12. 6.3
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QUIZ #7CR #7 due tomorrow
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The Quadratic FormulaHa ha...This one is pretty funny, right???
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Solve for x in the following 2 problems:
1. x2 5x 36 = 0
2. 2x2 + 2x = 12
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Often, the simplest way to solve "ax2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor.
But sometimes the quadratic is too messy, or it doesn't factor at all, or you just don't feel like factoring.
While factoring may not always be successful, the Quadratic Formula can always find the solution.
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For ax2 + bx + c = 0, the value of x is given by:
This formula is called The Quadratic Formula.
***Remember, a, b, and c represent the coefficients in our equation. Solving ax2 + bx + c = 0 for x means, among other things, that you are trying to find xintercepts.
Key Point: Sometimes they ask you to find the ROOTS. This is a fancy way of asking you to solve for the variable!
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http://www.youtube.com/watch?v=TVIcjaKt_A8&feature=fvwrel
http://www.youtube.com/watch?v=2lbABbfU6Zc
http://www.youtube.com/watch?v=jGJrH49Z2ZA&feature=related
There are some songs that will help you memorize the Quadratic Formula.
I use the song "Pop Goes The Weasel."
This is a cute little video that students at Westerville South High School in Ohio created. Maybe you guys could make one???
(Start at 3:00)
This one is pretty bad...
Nov 59:22 AM
Solve for x in each of the following in simplest radical form:
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Find the roots in each of the following equations in simplest radical form:
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Homework: p. 10, 11 #6, 8, and 16
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ANSWERS to p. 10 11 #6, 8, and 16:
6. x = 4,
8. x =
16. n =
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Quadratic Formula with Sum and Product Rules
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Remind me again how you would solve the following quadratic equation for the value of x in simplest radical form:
1.
Since this is an equation, there must be a way to check it. We could substitute our answer back into the original equation, but our answer is really messy. Plus there are 2 of them! There must be another method of check our
roots of the quadratic equation.
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Sum Rule:
Product Rule:
The method we will use to check our answers to a quadratic equation is called the
SUM RULE and
PRODUCT RULE
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Review