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Zeros of the polynomial function are the points where f(x) = 0. What values of x will zero out the equation? Answer: x= 5 and x= 6. These are also the x-intercepts.
The graph looks like this:


For your second question, A=P[1 +r/n]^nt is for interest compounded for periods such as annually, semiannually, quarterly, daily, etc. The formula A=Pe^rt is for interest compounded continuously.

Find the accumulated value of an investment of $10,000 at 4% compounded annually for 17yrs.
Use the first formula above, where P=$10,000, r=.04, n=1 time per year, and t= 17 years.

I get this with this formula and a calculator: A= 10000[1+.04)^17 =$19479.00496
Final answer: $19,479.00 What did you get??

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This is an EXCELLENT question!! When solving a rational equation, you are actually multiplying both sides of the EQUATION by the LCD. It is a property of equations that allows you to multiply both sides of the equation by the same non-zero number or expression.

However, when you are simplifying rational expressions--did you get that key word there?? EXPRESSIONS!! Now you are NOT dealing with an equation, but only an expression. So with an EXPRESSION, you don't have the properties of EQUATIONS!!

One of my favorite questions! Thanks for asking it!!

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Have you done anything with the laws of exponents? For example, when you divide with the same base number, you subtract exponents:


If it had been can you see that if you subtract exponents you would have which could be written ?

This same problem could also be simplified as a fraction to be reduced by dividing out the five x factors in the numerator with five of the x factors in the denominator. In this method, all the x factors in the numerator divide out, and you would be left with a 1 in the numerator since they all divided out and seven factors of x left in the denominator:


What you have just done is derived the formula

In summary, any time you have a negative exponent, you get a fraction whose numerator is 1, and whose denominator is the same as the original problem but with a positive exponent. For examples:




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Begin with


Next, what happens if you make that ?

Answer: It opens the graph up to a broader shape!

Now, what happens if you make the function negative

Answer: It inverts the graph!

Now what happens if you make that

Answer: It moves the graph 3 units to the right!

Finally, what happens if you subtract 3 from the previous function?

Answer: It moves the graph down three units.

Pretty neat, huh??

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D=RT
T=D/R

T(x) = D/R + D/R
T(x) = 200/x + 240/(x-20)

When x= 100,
T(100) = 200/100 + 240/(100-20)
T(100) = 200/100 + 240/80
T(100) = 2 + 3 = 5 hours

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Does not reduce. You can factor it if you like, but factoring is not usually necessary since the fraction does not reduce.


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Let x = speed of the current
15 +x = speed of the boat downstream (with the current)
15-x = speed of the boat upstream (against the current)
Distance upstream = Distance downstream = 36 miles

D=RT
T= D/R

Total Time = D/R + D/R = 5 hours


Multiply both sides of the equation by the LCD which is (15+x)(15-x):







x= 3 or x= -3

Reject the negative answer. x=3 mph = speed of the current.

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You are correct! And the trinomial does NOT refactor.

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, so



If , then

5(n+1) = 6n
5n+5 = 6n

Subtract 5n from each side:
5= n
Remember to check the value of n to make sure you don't have a log of a negative number. You do not, so the answer n=5 is acceptable.

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She spends $7 per week on sodas, which equals 1/4 of the total allowance.
Let x = total allowance

Here is the equation:

One fourth of total allowance equals $7.


Multiply both sides by 4:


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If the line contains the origin, that means that in the equation of a line
y=mx+b

the value of b = 0. All you need is the slope, which is rise over run, up 1 and over 2 to get from (0,0) to (2,1). The slope m is rise over run = 1/2, so the equation is
or

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Let me take a shot at it anyway. Is this what you mean:
???

If that is the question, then you can say that if , then .

Likewise, means that
, which is a quadratic equation. Get this equal to zero by adding -3x +1 to each side of the equation:


This does not factors so you'll have to solve by quadratic formula. Before I go to a lot of trouble to do that, let's check to see if I copied the problem correctly. You can repost this or send me an Email to finish the problem.

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Let D = distance home.

D=RT


Time = time riding + time walking = 2.25 hours


Multiply both sides by the LCD which is 32:



miles

Does that look right??

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I'm not sure I understand your question, but I'll try to answer it anyway.

When you write log with no base number, then it is assumed that you mean log base 10. Let's do a beginners lesson in logarithms.

log 10 means , which means "what power would you raise 10 to in order to get 10?" The answer is the power which is .

log 100 means , which means "what power would you raise 10 to in order to get 100?" The answer is the power which is .

log 1000 means , which means "what power would you raise 10 to in order to get 1000?" The answer is the power which is .

It so happens that if you have , then the answer is !








In your problem, I'm not sure what the power of 10 was, but whatever it was, I'll bet that power is the answer to the problem.

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Let x = cost.
.35x = profit

Cost + Profit = Selling price
x + .35x = 670
1.00 + .35x = 670
1.35x = 670

Divide both sides by 1.35:
x= 670/1.35= $496.30 approximately

It usually comes out more even than this, but such is life sometimes. I still think it is correct.

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Do you mean:


You need to factor this by finding two numbers whose product is 112 and whose difference is 9. If you can't think of the combination, start at the beginning with 1:

1*112
2* 56
3 does not go into 112
4* 28
7* 16

That's it! There is the difference of 9!


or
or

There are no real numbers that you can square to get a negative number, so has no real solutions. If you wanted complex solutions also, then the solutions of the last part would be

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Yes it is!

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P.S. I seem to have missed posting a lesson plan that might be helpful related to this question. I have it on my website if you care to look at it. Go to my website by clicking on my tutor name "rapaljer" anywhere in algebra.com, then click on "Basic Algebra", then "Samples from Basic Algebra One Step", then go to "Chapter 1", and look for "Like Terms and Distibutive Property". That section addresses a lot of terminology questions and a lot of small issues that we who know math take for granted. Here is the URL for this, in case you can click on the following and go there directly:

http://www2.scc-fl.edu/rrapalje/BasicAlgebra/Samples%20from%20Basic%20One%20Step%20Ch%201/1106%20Like%20Terms%20and%20Distrib%20Prop.htm.

I'll try to get it posted also to algebra.com as a lesson plan when I get a chance.

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PLEASE NOTE: I made a TERRIBLE error in this problem, now corrected thanks to the CORRECT solution by AnyltcPhil. Thanks Phil.



Notice that


Does that match up to any of your multiple choice answers?

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If Sally can paint the house in 4 hours, then in 1 hour she can paint 1/4 of the house.
If John can paint the house in 6 hours, then in 1 hour he can paint 1/6 of the house.
Let x = number of hours it would take them together to paint the house, then working together, in 1 hour they can paint 1/x of the house.

The equation is this:


Multiply both sides by the LCD which is 12x:



= 2.4 hours

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I would have to agree with the tutor, since I found two solutions when I solved it myself. I don't entirely understand your notation, but I think you may have lost a solution when you went from a quadratic equation to a linear equation:
(logx 3)^2 - 3logx 3 = 4
logx 3 - 3 = 4 / (logx 3)

It might have been safer if you had set it equal to zero, and factored it.
[ logx 3 - 4] [logx 3 + 1] = 0

This way, you too will get two solutions.

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Let x = amount of water evaporated
18 = amount of liquid before
18-x = amount of liquid after

Amount of salt before = Amount of salt after
.20*18= .50(18-x)
3.6 = 9.0 - .5x

Subtract 9.0 from each side
3.6 - 9.0 = 9.0 - 9.0 - .5x
-5.4 = -.5x

Divide by -.5:


x= 10.8 liters

Check:
.2* 18 = 3.6 before
.5*(18-10.8)= .5(7.2)= 3.6 after

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This is NOT an equation, so you can't solve it!

However, if you meant
x^2 -x - 30 = 0
(x-6)(x+5) = 0
x=6 or x= -5

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Just get the like terms together and combine like terms (remember that p means 1p):
-7n+3p -4n+p
-7n-4n +3p +1p
-11n + 4p

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I'd never thought of it before, but I suppose you could square each of the terms, then take twice the product of each combination of terms.



I guess that would do it, right?

Try it out:


=
=

I don't know! It probably is easier (and safer!) to just multiply it out the regular way. Maybe that's why I never heard of this "shortcut"!!

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You went to a lot of trouble to write this out. In the form of fractions is this what you meant?



Is that what you meant, or perhaps either you or I mis-copied something. At least I hope we mis-copied something, because I've looked at this, and it doesn't simplify. I see a couple of places it would factor (the first numerator and both denominators factor), but I don't see how it will simplify!!

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If the equation of a line is given in "standard form", such as Ax+By=C, then you must solve for y. For example, given 3x+2y = -12, you begin by subtracting 3x from both sides:

3x+2y=-12
2y = -3x - 12

Next, divide both sides by the coefficient of y, which is 2:



This is called slope-intercept form of the equation since the coefficient of x is the slope (in this case ), and the y intercept is the number term (in this case the y intercept is -6).

I'm assuming that this is what you meant by "slope format"

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Do you mean

If so, then cross multiply (or equivalently multiply both sides by the denominator (3x-5) ).



This is quadratic, so set equal to zero by subtracting 2 from both sides:


Fortunately the trinomial factors (or you can use the quadratic formula!):


Set each factor equal to zero:
3x+1=0
3x=-1
x=-1/3

x-2= 0
x=2

Neither answer makes a denominator zero, so both answers are acceptable.

On the other hand suppose the solution were intended to be


Now the LCD is 3x, so multiply both sides by 3x:




This would best be solved by the quadratic formula , since it does not factor:

Ugly, but that's life!

Then again, maybe the problem was intended to be . Now the LCD is just 3, so you can multiply both sides of the equation by 3:



Subtract 2x from each side:



Now that's better!! Do you see why it is SO important to state your problems clearly? Remember that EVERYONE at algebra.com uses the same order of operations agreement (including students, right?), and it really helps if in doubt to put parentheses around denominators and quantities inside radicals, etc. That way we ALL know what you mean in the problem.

As you can see, this one problem had THREE interpretations, and THREE different answers!! Help us out by stating your problem clearly!!

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I would multiply both sides of the equation by the denominator, which is 6, to clear the fraction.
|4x-8| = 36x

That is the FIRST thing I would do. Did you by any chance want to see the second and third thing to do, or was that your only question?

Next, remember that there are two solutions for an absolute value problem like this. Also, you will need to check the answers to make sure you do not accidentally get an absolute value equal to a negative. If this happens you must reject the answer. You must solve:

FIRST SOLUTION:
4x-8 = 36x

Subtract 4x from each side:
-8 = 32x

Divide both sides by 32:
x= -8/32 = -1/4

This answer must be rejected, since in the original problem, it causes an abolute value to equal a negative, which is not allowed.

SECOND SOLUTION:
4x-8 = -36x

Subtract 4x from each side:
-8=-40x

Divide both sides by -40
x=-8/-40 = 1/5

The final answer should be x= 1/5 or x = 0.2

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Set equation equal to zero by adding to each side:


Factor the common factor of 2:


Factor the trinomial:


Set each factor equal to zero. The 2 from the common factor does not give any solutions.
x-2=0, so x=2

x+1=0, so x=-1

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, where t=0 in 1990.

The problem is to find p(t) in year 2000 when t=10. In other words, find p(10).


= 29,045,856, using a calculator.

Since the problem is only given to 2 significant digits of accuracy, probably the answer is not more accurate than this. Call it about 30,000,000. I'm not sure what groundrules your instructor or the textbook may have given concerning accuracy of the problem.

For more information on this topic, especially on finding the growth constant, see my lesson plan in algebra.com in "logarithms" topic on Population Growth.

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This is correct, except for a couple of equal signs that should have been plus signs. I fixed those for you. Now, since this is a fractional equation, you have to check the answers to make sure you didn't accidentally divide by zero. There are certain values of x that are NOT acceptable. In this problem, x cannot equal -4, so you have to reject this answer, since it makes a denominator zero--NOT allowed! The answer is NO SOLUTION!

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