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27m^8-90m^6+45m^4

The first step in ANY factoring problem is to take out the COMMON FACTOR. In this case, it would be 9m^4:

9m^4(3m^4 -10m^2 + 5)

The resulting trinomial does not seem to factor

I have a LOT of resources on my own website on the topic of FACTORING. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in "Chapter 2" for various sections on factoring, especially "Section 2.03, Factoring the Common Factor." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

In addition, I have a video of me teaching this topic in the classroom a few years ago. To see this FREE video, from my Homepage, look for "Rapalje Videos in Living Color". Select "Basic Algebra", and look for "FACTORING."

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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The period of y=sin x is normally 2pi. Therefore, y=sin3x is 2pi/3. Of course the 4 is the amplitude, which is irrelevant to the period.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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FALSE! Tan pi/2 = undefined!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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S= a + ar + ar^2 + ar^3 + ... + ar^(n-1)(The series stops at n-1 power because there are n terms, but the first one is not multiplied by r)
rS=ar +ar^2 + ar^3 + . . . + ar^n

Subtract the above equations. All the middle terms subtract out!
S-rS = a - ar^n

Factor out the S on the left and the a on the right side of the equation:
S(1-r) = a(1-r^n)

Divide both sides by (1-r):


This does not agree exactly with your final answer, so multiply both numerator and denominator by -1, and it will give you the final result you are looking for!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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TRUE!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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When you get 0=0, this means that the statement is ALWAYS TRUE, so the lines are actually the SAME LINE. There are infinitely many solutions.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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I have a LOT of resources on my own website on the topic of GRAPHING and SLOPE of a LINE. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in "Chapter 4" for "Section 4.02, Slope Intercept Method." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

In addition, I have a video of me teaching this topic in the classroom a few years ago. To see this FREE video, from my Homepage, look for "Rapalje Videos in Living Color". Select "Basic Algebra", and look for "Graphing and Slope of a Line."

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

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This is a FACTORING problem. The first step in any factoring problem is to take out the COMMON FACTOR. In this case, the common factor is a

a^4-ab^3
a(a^3-b^3)

This is a DIFFERENCE of CUBES, which requires a special formula that is actually the product of a BINOMIAL times a TRINOMIAL.
a(______)(_________)

The binomial is the first, which is "a" minus the second, which is "b":
a(a-b)(___________)

The trinomial consists of the square of the first, PLUS the product of the first times the second, plus the square of the second.
a(a-b)(a^2+ab+b^2)

I have a LOT of resources on my own website on the topic of FACTORING. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Intermediate Algebra", and look in "Chapter 2" for "Section 2.01, Factoring." Select the Section called "Sum and Difference of Cubes." It's a topic that you probably will need more help with than other topics. This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

In addition, I have a video of me teaching this topic in the classroom a few years ago. To see this FREE video, from my Homepage, look for "Rapalje Videos in Living Color". Select "Intermediate Algebra", and look for "Factoring". You don't have to watch the whole video, which is over an hour long, but instead you can fast forward the video so you can see the topic that you need help with.

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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Do you mean this?


If this is it, then factor everything that you can:


Divide out the (m+1) factors and all the (m-1) factors, and what is left over is .

I have a chapter on fractions on my own website. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in "Chapter 3" for "Section 3.02, Multiplying and Dividing Fractions." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

You can put this solution on YOUR website!
If (x-1/x)=5, then square both sides and get this:



Add +2 to each side:


Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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The first step in factoring is to try to take out any common factors. In this case, you have a common factor of a, so take out the "a" factor.

a^4-ab^3
a(a^3 -b^3)


Now, this is a DIFFERENCE OF CUBES. There is a special formula for this in the form of a BINOMIAL times a TRINOMIAL. You may want to see my website for additional explanation of this formula. See the note at the end of this problem.
a(___)(_________)
a(a-b)(________)
a(a-b)(a^2+ab+b^2)

To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "MATH IN LIVING COLOR". Choose "Intermediate Algebra", and look in "Chapter 2" for "Section 2.01, Factoring." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

In addition, I have a video of me teaching this topic in the classroom a few years ago. To see this FREE video, from my Homepage, look for "Rapalje Videos in Living Color". Select "Intermediate Algebra", and look for "Factoring."

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

You can put this solution on YOUR website!
Plug in x=-1.
y=-x+7
y=-(-1)+7
y=1+7
y=8

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

You can put this solution on YOUR website!
This is a problem called "Rationalizing a BINOMIAL Denominator!" To rationalize a denominator means to "get rid of the radicals in the denominator." In the case of a binomial denominator, it takes a special trick to do what needs to be done here. You have to multiply the numerator and denominator by what is called the "CONJUGATE" of the denominator. That is, the SAME as the denominator, but with the OPPOSITE sign in the middle!! We do this for one reason--because it WORKS!! In this case, you have to multiply the numerator and denominator by . It looks like this:


*

Now, multiply the numerator and denominator out. Notice that the denominator is left with NO RADICALS!


In the denominator, notice that the middle terms subtract out, and the other denominator terms are PERFECT SQUARES! Simplify the numberator!




Factor the numerator so you can reduce the fraction:




FINAL ANSWER

I have a LOT of resources on my own website on the topic of SQUARE ROOTS. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "MATH IN LIVING COLOR." For problems like this one, please see "Intermediate Algebra" and look in "Chapter 3" for "Section 3.04 Rationalizing Denominators." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

In addition, I have a video of me teaching this topic in the classroom a few years ago. To see this FREE video, from my Homepage, look for "Rapalje Videos in Living Color". Select "Intermediate Algebra", and look for "Radicals and Fractional Exponents".

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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This is a LOT easier than it looks! All you have to do is realize that (1,y) means that x=1, and you have to find the value of y!

If y=-x +7, just plug in the value of x=1, and find y:
y=-1+7
y=6

That's it! FINAL ANSWER!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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The quadratic formula is

In this case, x^2 +12x+35=0, a=1, b=12, and c=35.





or
or

This problem can be checked by solving by FACTORING:
x^2 +12x+35=0
(x+5)(x+7)=0
x=-5 or x=-7

I have a LOT of resources on my own website on the topic of QUADRATIC FORMULA. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Intermediate Algebra", and look in "Chapter 4" for "Section 4.03, Quadratic Formula." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

You can put this solution on YOUR website!
No, I'm afraid you missed this concept!
The second question is easiest, so let's do it first.

You have to do the square roots first, and THEN you add the results together, but ONLY IF YOU HAVE LIKE TERMS or LIKE RADICALS!!

FINAL ANSWER for the second part of your question!

Now, the FIRST part of your question is much harder!! In fact, I don't know how they could have made this problem any harder for you! (If you don't understand my explanation here, PLEASE see the note at the end of this problem telling you about my website which has some MUCH EASIER problems to help you get started on this concept!!)



The first step is to SIMPLIFY each square root SEPARATELY! You have to try to find a perfect square that is a factor of 486. It just so happens that the PERFECT SQUARE 81 divides evenly into 486. It goes 6 times, so you can write
486=81*6.

Now do the same thing for 384. Find a perfect square that goes evenly into 384. I can't think of a perfect square that goes into this, but the previous part, square root of 486, came out with a leftover factor of 6. Do you think maybe 6 goes into 384? Indeed it does. It goes 64 times, which just happens to be a perfect square!! You can write
384=64*6

(I told you this was a HARD ONE!!)
NOW,





Now, DO the square roots that you know how to do, and leave the other ones alone!



Now, notice that you have both square roots EXACTLY the same!! This means that you can COMBINE LIKE TERMS! Add the 9 + 16, and you get 25!


I warned you that this was a HARD problem!! You probably need to see my own website in order to get some EASIER problems before you can really understand this one! I have a LOT of resources on my own website on the topic of SQUARE ROOTS. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in "Chapter 5" for "Section 5.01, Square Roots" and "Section 5.03 Combining Like Radicals." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

In addition, I have a video of me teaching this topic in the classroom a few years ago. To see this FREE video, from my Homepage, look for "Rapalje Videos in Living Color". Select "Basic Algebra", and look for "Radicals (Square Roots)".

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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The first step in simplifying a square root is to SORT it into TWO square roots where the first square root contains only perfect squares, and the second square root has all the "left-over" factors. Remember also that if you have a variable raised to a power, if the power is EVEN, like x^4 or y^8, then this IS a perfect square, and to take the square root, you take HALF of the exponent.

Okay? Here we go:




Now, take the square root of the first one, since it IS a perfect square. Remember, to take HALF of the exponents! However, the second square root is NOT a perfect square, so just leave it like it is!


I have a LOT of resources on my own website on the topic of SQUARE ROOTS. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in "Chapter 5" for "Section 5.01, Square Roots." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

In addition, I have a video of me teaching this topic in the classroom a few years ago. To see this FREE video, from my Homepage, look for "Rapalje Videos in Living Color". Select "Basic Algebra", and look for "Radicals (Square Roots)".

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

You can put this solution on YOUR website!
First, you must set this equation equal to zero!
5x^2-10x =0

Now factor the left side. There is a common factor of 5x:
5x(x-2)=0

There are two solutions.
5x= 0
x=0

OR x-2=0
x=2

I have a LOT of resources on my own website on the topics of FACTORING and SOLVING QUADRATIC EQUATIONS. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in "Chapter 2 for the appropriate sections. This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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Let x = number of hours to fill the pool with both draining and filling.

1/6 = the part of the pool that can be filled in 1 hour
1/8= the part of the pool that can be drained in 1 hour

1/x= the part of the pool that is filled with both pipes open

1/6-1/8=1/x

Multiply both sides of the equation by 24x:


hours

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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When the equation is in slope-intercept form, y=mx+b, the slope is always the coefficient of x, and the y-intercept is always the number term.

y=x+2
Slope = 1, y-intercept = 2.

y=4/5x+2
Slope = 4/5, y-int = 2

y=7/4x+2
Slope = 7/4, y-int = 2

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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To put this in slope intercept form, you have to solve for y. Start by subtracting 5x from each side.

5x-7y=9

-7y=-5x+9

Now divide by -7:



Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

You can put this solution on YOUR website!
Is there a diagram or picture that we are not seeing here?

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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Graph y=x^2 -2x-8 with a calculator.


As you can see, the x-intercepts are at x=-2 and x=4.

This can also be determined algebraically by letting y=0 and solving
0=x^2-2x-8
0=(x-4)(x+2)
x=4 or x=-2

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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-4, -3, -2, -1, 0, 1

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This is a 3-dimensional graph, with x, y, and z axes. This point will be on the y axis at a distance of 4 units from the origin. I can't draw this on the algebra.com format.

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The slope is the coefficient of x, which is 2.

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Let x = length of the body, since that is at the end of the sentence.
Now, the tail = length of the head plus half the length of the body.
tail = 10 + 1/2*x

The equation is that the body = head + tail
x= 10 + 10+ 1/2x

Subtract 1/2x from each side:
1/2x= 20

Multiply both sides by 2:
x=40 inches = Body
Tail = 10 + 1/2x
Tail = 10 + 20 = 30 inches

Total fish = Head + Body + Tail
Total fish = 10+40+30 = 80 inches! (BIG FISH!!)

Check:
Body = Head + Tail
40=10+30

It checks!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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Multiply both sides by 2:



Factor the x:

There are two solutions:
x=0 NOTE: THIS ANSWER MUST BE REJECTED, SINCE log(0) is UNDEFINED!!

6x-1=0
6x=1
x=1/6 This one seems to work!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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If the equation is y=ax^2 +bx +c, then the vertex is always at the value where x=-b/(2a). After you find x, substitute this back into the original equation (or let the calcluator do it for you!!) to find y.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

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Let x= the 2nd number. (I always like to let x = the quantity at the end of the sentence or phrase!)

Now, the first number is 4 more than twice the second number, where the second number is x.

Therefore, first number = 2x+4.

Now the equation is that the total (or sum!) of the numbers is 25.
x+2x+4=25

You said that you can solve it, so go for it!

I have a LOT of resources on my own website on the topic of WORD PROBLEMS. To find my website, do a "BING" or a "GOOGLE" search for my last name "Rapalje". Near the top of the search list, you should see "Rapalje Homepage." Click on this, and look near the top of my homepage for the link "Basic, Intermediate and College Algebra: One Step at a Time." Choose "Basic Algebra", and look in "Chapter 1" for "Sections 1.09 and 1.10, Word Problems." This is my own explanation especially written for students who have trouble understanding math. I wrote this to students, NOT mathematicians, and I think you will find it easier to understand than your own traditional textbooks!! In addition to this page, the entire website is supported by my "MATH IN LIVING COLOR" pages, where the hardest problems are solved IN COLOR.

If you like my website, please recommend it to family and friends who have trouble understanding math!! It's ALL FREE!!!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

You can put this solution on YOUR website!
Not enough information is given to solve this one. The area could be 30 square inches because it is 5 by 6 or it could be 3 by 10, or many different possiblities.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus