Solving Equations By Factoring 6-8
Zero Products Property -- a "friend" you can count on..."
For any two rational numbers a and b, if ab = 0, then either a = 0 or b = 0 or both equal zero
How does this help you?
Try solving
x2 + 10x + 24 = 0
Your first reaction is to subtract 24 from both sides you would get
x2 + 10x = -24 BUT NOW WHAT?????
It's a quadratic ( x2 term). You can't isolate x because there is another term with the x2 in it. THe Zero Products Property will help you solve quadratics.
Using this property we can solve quadratic equations by factoring the equation and setting each factor to zero and solving them.
x2 + 10x + 24 = 0
factor as usual-- using your rules from the other sections:
(x +6) (x +4) = 0
set each one to zero
x + 6 = 0 and x + 4 = 0
so x = -6 and x = -4
Plug them back-- and it works!!
IF you are already given it factored--just set each to zero
(5x +1)(x-7) = 0
Using the Zero Products Property we know either 5x - 1 must be equal to zero or x - 7 must equal zero... that's the only way that the product could be zero...
Set both equal to zero and solve.
if 5x + 1 = 0 then x = -1/5 ( its just a 2 step equation)
if x - 7 = 0 then x = 7
If you substitute these answers for x back in the original equation, they will both end up as 0 = 0 which is what you want. The left side equals the right side!!
In the second example, the equation was already factored.
WIth the first example, you saw you needed to first factor the equation ( if possible) and then set each factor to zero to solve.
MAKE SURE YOU MOVE EVERYTHING TO ONE SIDE OF THE EQUATION. YOU MUST HAVE ZERO ON ONE SIDE OF THE EQUATION TO USE THE ZERO PRODUCTS PROPERTY
x2 = 16
First move the 16 to the left side
x2 -16 = 0
Now... you have the difference of two squares
(x + 4)(x-4) = 0
Set each factor to zero and solve
x + 4 = 0 and x - 4 = 0
so x = -4 and x = 4
Why do you have to have zero on one side ?
NEW ALGEBRA TERM
root-- any solution that turns the equation into the value of zero is called a root of the polynomial, or a ZERO of the polynomial because when you are graphing the polynomial, the y value is zero at this point.
So if the directions say " FInd the roots of..." it just means get zero on one side of th quation, factor, set each factor to zero and solve.
Summary of solving quadratics
solutions = roots= eros of the polynomials
1. Get zero on one side of the equation
2. Factor
3. Set each piece ( factor) equal to zero
4. Solve as one or two step equations
5. Check by substituting in the ORIGINAL equation
Wednesday, January 12, 2011
Tuesday, January 11, 2011
Math 6 Honors (Period 6 and 7)
Congruent Figures 4-7
Two figures are congruent if they have the same size and same shape. If we could lift on of the figures and place it directly on top of the other... all three vertices would match up with the others. In the example in our book ( page 132) we have two congruent triangles ∆ABC and ∆XYZ A would fall on X, B would fall on Y and C would fall on Z. These matching vertices are called corresponding vertices. Angles at corresponding vertices are corresponding angles and the sides joining corresponding vertices are corresponding sides.
The book states Corresponding angles of congruent figures are congruent.
and
Corresponding sides of congruent figures are congruent.
In class we discussed how to abbreviate the above -- when dealing with triangles.
CPCTC
Corresponding PARTS of congruent triangles are congruent!!
When we name two congruent figures we list corresponding vertices in the same order.
∆ABC ≅ ∆XYZ or ∆CAB ≅ ∆ZXY or ∆BCA ≅ ∆YZX
we know that
∠A ≅ ∠X and ∠B ≅ ∠Y and ∠C ≅ ∠Z
and the segments ( which are denoted with a line (but w/o arrows)above each of the two letters
AB ≅ XY and BC ≅ YZ and CA ≅ ZX
If two figures are congruent, we can make the coincide -- occupy the same place-- by using one or more of the following basic rigid motions:
Translation or slide
Rotation
Reflection or flip or mirror
Check the book on page 133 for good examples of these three rigid motions... I like to think of Tetris moves!!
Two figures are congruent if they have the same size and same shape. If we could lift on of the figures and place it directly on top of the other... all three vertices would match up with the others. In the example in our book ( page 132) we have two congruent triangles ∆ABC and ∆XYZ A would fall on X, B would fall on Y and C would fall on Z. These matching vertices are called corresponding vertices. Angles at corresponding vertices are corresponding angles and the sides joining corresponding vertices are corresponding sides.
The book states Corresponding angles of congruent figures are congruent.
and
Corresponding sides of congruent figures are congruent.
In class we discussed how to abbreviate the above -- when dealing with triangles.
CPCTC
Corresponding PARTS of congruent triangles are congruent!!
When we name two congruent figures we list corresponding vertices in the same order.
∆ABC ≅ ∆XYZ or ∆CAB ≅ ∆ZXY or ∆BCA ≅ ∆YZX
we know that
∠A ≅ ∠X and ∠B ≅ ∠Y and ∠C ≅ ∠Z
and the segments ( which are denoted with a line (but w/o arrows)above each of the two letters
AB ≅ XY and BC ≅ YZ and CA ≅ ZX
If two figures are congruent, we can make the coincide -- occupy the same place-- by using one or more of the following basic rigid motions:
Translation or slide
Rotation
Reflection or flip or mirror
Check the book on page 133 for good examples of these three rigid motions... I like to think of Tetris moves!!
Pre Algebra (Period 2 & 4)
Comparing and Ordering Fractions
5-1
HOW TO FIND THE LEAST COMMON MULTIPLE:
LCM = Smallest number that your numbers can go into.
Just like GCF, let's look at the letters backwards to understand it!
Multiple = each number given in the problem must go into this number (example for multiples of 2: 2, 4, 6, 8.... multiples of 3: 3, 6, 9, 12... multiples of 5: 5, 10, 15, 20...)
Common = must be a number that ALL THE NUMBERS go into
Least = must be the SMALLEST number that ALL THE NUMBERS go into
There are the same ways to find it as the GCF:
1) List multiples of each number and circle the smallest one that is common to all the numbers ---> most of the time takes WAY TOO long!!
2) Circle every factor in the prime factorizations of each number that is different and multiply
3) List the EXPANDED FORM prime factorizations in a table and bring down ONE OF EACH COLUMN. Then multiply. (or you can do this with exponential form but you need to bring down the HIGHEST POWER of each column).
4) Using the BOX method from class create an L from the left side and the bottom row of relatively prime factors. It is their product.
THE DIFFERENCE BETWEEN GCF AND LCM:
For the GCF, you need the LEAST POWER of only the COMMON FACTORS.
For the LCM, you need the GREATEST POWER of EVERY FACTOR.
WHY DO WE NEED EVERY FACTOR THIS TIME?
Because it's a multiple of all your numbers!
Multiples start with each number, so all the factors that make up each number have to be in this common multiple of all the numbers.
For example, say we're finding the LCM of 12 and 15, that multiple must be a multiple of 12: 12, 24, etc.
AND 15: 15, 30, etc.
So the COMMON multiple must include 12 (2x2x3) AND 15 (3x5).
The LCM must have two 2s and one 3 or 12 won't go into it.
It must also have that same 3 that 12 needs and one 5 or it won't be a multiple of 15.
EXAMPLE: 54 and 36
LIST MULTIPLES OF EACH NUMBER UNTIL YOU SEE ONE THAT BOTH OF THEM GO INTO (the LEAST MULTIPLE that is COMMON to both)
36 = 36, 72, 108, 144 ...
54 = 54, 108
This more difficult than the listing method for GCF for 2 reasons: You don't know where to stop as you least the first number...multiples go on forever! and the numbers get big very fast because they're MULTIPLES, not factors!
PRIME FACTORIZATION METHOD
54 = 2x3x3x3
36 = 2x2x3x3
The LCM will be one of each factor of each number (but don't double count a factor that is common to both numbers...once you have it in your LCM, check it off in the other number if it's common...you don't need that factor again)
LCM = 2x2x3x3x3 = 108
Let's look at this calculation and think about why it works:
Why does it need TWO 2s?
Although 54 only needs ONE, 36 needs TWO or 36 won't go into the LCM.
Try it putting in only ONE 2: 2x3x3x3 = 54. Does 36 go into 54? NO
Why does the LCM need THREE 3s? Although 36 needs only TWO 3s, 54 needs THREE. Try it putting in only TWO: 2x2x3x3 = 36.
36 is TOO SMALL to even be a MULTIPLE of 54!
WHAT IF BY MISTAKE YOU "DOUBLE" UP FACTORS AND USE ALL OF THEM?
For the GCF, you would have got a number way too big to be a factor of the numbers. For the LCM, you will STILL GET A COMMON MULTIPLE! But it WON'T BE THE LEAST!
In fact if you double up, generally you'll get a really big number and the bigger the number is, the harder it is to use.
For example, if you use all the factors of both 36 and 54, you're just multiplying 36 x 54 = ????
1944!!!!
Would you rather use 108 or 1944???
LCM WITH A COLUMN APPROACH: You place the prime factorization for each number in columns like we did for the GCF, matching the factors in each column. If a factor doesn't match, it gets a separate column.
YOU'LL JUST TAKE ONE OF EACH COLUMN AND MULTIPLY! SAME EXAMPLE: 54, 36
36 = 2 x 2 x 3 x 3
54 = 2 x 3 x 3 x 3
LCM = 2 x 2 x 3 x 3 x 3
= 108
You can do it in exponential form, too.
If you do it in exponential form, you take the HIGHEST POWER of each column!
Works with variables the exact same way!
Take the HIGHEST power of EVERY variable (not just the ones that are common like the GCF)
YOU CAN FIND BOTH THE GCF AND THE LCM IN THIS SAME COLUMN FORMAT!
CHECKING THE LCM TO MAKE SURE IT WORKS:
Ordering or comparing fractions:(last method)
4: Make them into decimals because DECIMALS = FRACTION posers!
(or decimals are just fraction wannabes!)
Today, we will change fractions to decimals and decimals to fractions.
How to change a fraction to a decimal 1.
Divide (ALWAYS WORKS!)
EXAMPLE: 3/4 = 3 divided by 4 = .75
If the quotient starts repeating, then put a bar over the number(s) that repeat. OR
2. Use equivalent fractions (SOMETIMES WORKS!)
Works if the denominator can be easily made into a power of 10
SAME EXAMPLE: but this time you will multiply by 25/25 to get 75/100 = .75
3. MEMORY! Some equivalencies you should just know! EXAMPLE: 1/2 = .5
IF IT'S A MIXED NUMBER, JUST ADD THE WHOLE NUMBER AT THE END!
EXAMPLE: 8 3/4
For the fraction: 3 divided by 4 = .75
Add the whole number: 8.75
IF THE MIXED NUMBER OR FRACTION IS NEGATIVE, SO IS THE DECIMAL!
CHANGING TERMINATING DECIMALS TO FRACTIONS: EASY!!!
Read it, Write it, Simplify!
EXAMPLE:
Change .24 to a fraction
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
EXAMPLE with whole number:
Change 7.24 to a fraction
The 7 is the whole number in the mixed number so you just put the 7 at the end
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
4) 7 6/25
HOW TO CHANGE REPEATING DECIMALS TO FRACTIONS---> we will do that 2nd semester
HOW TO FIND THE LEAST COMMON MULTIPLE:
LCM = Smallest number that your numbers can go into.
Just like GCF, let's look at the letters backwards to understand it!
Multiple = each number given in the problem must go into this number (example for multiples of 2: 2, 4, 6, 8.... multiples of 3: 3, 6, 9, 12... multiples of 5: 5, 10, 15, 20...)
Common = must be a number that ALL THE NUMBERS go into
Least = must be the SMALLEST number that ALL THE NUMBERS go into
There are the same ways to find it as the GCF:
1) List multiples of each number and circle the smallest one that is common to all the numbers ---> most of the time takes WAY TOO long!!
2) Circle every factor in the prime factorizations of each number that is different and multiply
3) List the EXPANDED FORM prime factorizations in a table and bring down ONE OF EACH COLUMN. Then multiply. (or you can do this with exponential form but you need to bring down the HIGHEST POWER of each column).
4) Using the BOX method from class create an L from the left side and the bottom row of relatively prime factors. It is their product.
THE DIFFERENCE BETWEEN GCF AND LCM:
For the GCF, you need the LEAST POWER of only the COMMON FACTORS.
For the LCM, you need the GREATEST POWER of EVERY FACTOR.
WHY DO WE NEED EVERY FACTOR THIS TIME?
Because it's a multiple of all your numbers!
Multiples start with each number, so all the factors that make up each number have to be in this common multiple of all the numbers.
For example, say we're finding the LCM of 12 and 15, that multiple must be a multiple of 12: 12, 24, etc.
AND 15: 15, 30, etc.
So the COMMON multiple must include 12 (2x2x3) AND 15 (3x5).
The LCM must have two 2s and one 3 or 12 won't go into it.
It must also have that same 3 that 12 needs and one 5 or it won't be a multiple of 15.
EXAMPLE: 54 and 36
LIST MULTIPLES OF EACH NUMBER UNTIL YOU SEE ONE THAT BOTH OF THEM GO INTO (the LEAST MULTIPLE that is COMMON to both)
36 = 36, 72, 108, 144 ...
54 = 54, 108
This more difficult than the listing method for GCF for 2 reasons: You don't know where to stop as you least the first number...multiples go on forever! and the numbers get big very fast because they're MULTIPLES, not factors!
PRIME FACTORIZATION METHOD
54 = 2x3x3x3
36 = 2x2x3x3
The LCM will be one of each factor of each number (but don't double count a factor that is common to both numbers...once you have it in your LCM, check it off in the other number if it's common...you don't need that factor again)
LCM = 2x2x3x3x3 = 108
Let's look at this calculation and think about why it works:
Why does it need TWO 2s?
Although 54 only needs ONE, 36 needs TWO or 36 won't go into the LCM.
Try it putting in only ONE 2: 2x3x3x3 = 54. Does 36 go into 54? NO
Why does the LCM need THREE 3s? Although 36 needs only TWO 3s, 54 needs THREE. Try it putting in only TWO: 2x2x3x3 = 36.
36 is TOO SMALL to even be a MULTIPLE of 54!
WHAT IF BY MISTAKE YOU "DOUBLE" UP FACTORS AND USE ALL OF THEM?
For the GCF, you would have got a number way too big to be a factor of the numbers. For the LCM, you will STILL GET A COMMON MULTIPLE! But it WON'T BE THE LEAST!
In fact if you double up, generally you'll get a really big number and the bigger the number is, the harder it is to use.
For example, if you use all the factors of both 36 and 54, you're just multiplying 36 x 54 = ????
1944!!!!
Would you rather use 108 or 1944???
LCM WITH A COLUMN APPROACH: You place the prime factorization for each number in columns like we did for the GCF, matching the factors in each column. If a factor doesn't match, it gets a separate column.
YOU'LL JUST TAKE ONE OF EACH COLUMN AND MULTIPLY! SAME EXAMPLE: 54, 36
36 = 2 x 2 x 3 x 3
54 = 2 x 3 x 3 x 3
LCM = 2 x 2 x 3 x 3 x 3
= 108
You can do it in exponential form, too.
If you do it in exponential form, you take the HIGHEST POWER of each column!
Works with variables the exact same way!
Take the HIGHEST power of EVERY variable (not just the ones that are common like the GCF)
YOU CAN FIND BOTH THE GCF AND THE LCM IN THIS SAME COLUMN FORMAT!
CHECKING THE LCM TO MAKE SURE IT WORKS:
Ordering or comparing fractions:(last method)
4: Make them into decimals because DECIMALS = FRACTION posers!
(or decimals are just fraction wannabes!)
Today, we will change fractions to decimals and decimals to fractions.
How to change a fraction to a decimal 1.
Divide (ALWAYS WORKS!)
EXAMPLE: 3/4 = 3 divided by 4 = .75
If the quotient starts repeating, then put a bar over the number(s) that repeat. OR
2. Use equivalent fractions (SOMETIMES WORKS!)
Works if the denominator can be easily made into a power of 10
SAME EXAMPLE: but this time you will multiply by 25/25 to get 75/100 = .75
3. MEMORY! Some equivalencies you should just know! EXAMPLE: 1/2 = .5
IF IT'S A MIXED NUMBER, JUST ADD THE WHOLE NUMBER AT THE END!
EXAMPLE: 8 3/4
For the fraction: 3 divided by 4 = .75
Add the whole number: 8.75
IF THE MIXED NUMBER OR FRACTION IS NEGATIVE, SO IS THE DECIMAL!
CHANGING TERMINATING DECIMALS TO FRACTIONS: EASY!!!
Read it, Write it, Simplify!
EXAMPLE:
Change .24 to a fraction
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
EXAMPLE with whole number:
Change 7.24 to a fraction
The 7 is the whole number in the mixed number so you just put the 7 at the end
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
4) 7 6/25
HOW TO CHANGE REPEATING DECIMALS TO FRACTIONS---> we will do that 2nd semester
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