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a) Correct.

b)

Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So the max profit is $6,470.

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Are you sure that the equation isn't ??


Start with the given expression


Group like terms


Factor out the GCF out of the first group. Factor out the GCF out of the second group


Since we have the common term , we can combine like terms


So factors to

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Let's see if and are solutions to the first equation.


Start with the first equation.


Plug in and .


Multiply


Combine like terms.


Since the equation is true, this means that (5,2) is a solution to the first equation.


Let's see if (5,2) is a solution to the second equation.


Start with the second equation.


Plug in and .


Multiply


Combine like terms.


Since the equation is true, this means that (5,2) is also a solution to the second equation.


Since all of the equations of the system are true, this means that (5,2) is a solution to the given system.

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3)

Start with the given expression


Now take half of the x coefficient and square it to get . This is the value that you add to the expression to make it a perfect square.

So is a perfect square.

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4)

Is the equation ???



Start with the given equation.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Simplify the square root


Break up the fraction.


Reduce.


or Break up the expression.


So the answers are or


which approximate to or


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OR....

Is the equation ???




Start with the given equation.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Simplify the square root


Break up the fraction.


Reduce.


or Break up the expression.


So the answers are or


which approximate to or

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Start with the given expression.


Multiply the fraction by .


Combine the fractions.


FOIL the numerator.


FOIL the denominator.


Multiply.


Combine like terms.


Break up the fraction.


So .


So the expression is now in standard form where and

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Start with the given equation.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the answers are or

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Start with the given expression.


Rearrange the terms.


Looking at we can see that the first term is and the last term is where the coefficients are 1 and -45 respectively.

Now multiply the first coefficient 1 and the last coefficient -45 to get -45. Now what two numbers multiply to -45 and add to the middle coefficient 4? Let's list all of the factors of -45:



Factors of -45:
1,3,5,9,15,45

-1,-3,-5,-9,-15,-45 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -45
(1)*(-45)
(3)*(-15)
(5)*(-9)
(-1)*(45)
(-3)*(15)
(-5)*(9)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4

First Number	Second Number	Sum
1	-45	1+(-45)=-44
3	-15	3+(-15)=-12
5	-9	5+(-9)=-4
-1	45	-1+45=44
-3	15	-3+15=12
-5	9	-5+9=4

From this list we can see that -5 and 9 add up to 4 and multiply to -45


Now looking at the expression , replace with (notice adds up to . So it is equivalent to )




Now let's factor by grouping:


Group like terms


Factor out the GCF of out of the first group. Factor out the GCF of out of the second group


Since we have a common term of , we can combine like terms

So factors to


So this also means that factors to (since is equivalent to )



------------------------------------------------------------



Answer:
So completely factors to

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Start with the given expression


Factor out the GCF


Group like terms


Factor out the GCF out of the first group. Factor out the GCF out of the second group


Since we have the common term , we can combine like terms


Factor to get


So completely factors to

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I'd start with the left side and work my way to get it equal to the right side (if you go with the left side, do NOT change the right).


Recall the identities:

cos(A+B) = cos(A)cos(B)-sin(A)sin(B), and


cos(A-B) = cos(A)cos(B)+sin(A)sin(B)


--------------------------------------------------



cos(A+B)+cos(A-B)=2cos(A)cos(B) ... Start with the given equation.


[cos(A)cos(B)-sin(A)sin(B)]+cos(A-B)=2cos(A)cos(B) ... Use the first identity given above


[cos(A)cos(B)-sin(A)sin(B)]+[cos(A)cos(B)+sin(A)sin(B)]=2cos(A)cos(B) ... Use the second identity given above


[cos(A)cos(B)+cos(A)cos(B)]+[-sin(A)sin(B)+sin(A)sin(B)]=2cos(A)cos(B) ... Rearrange the terms.


2cos(A)cos(B)+0=2cos(A)cos(B) ... Combine like terms.


2cos(A)cos(B)=2cos(A)cos(B) ... Simplify


Since both sides are equal, we've verified the identity.

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Since , this means that





or


or


or


or


etc...

----------------------------------

So for every increment of the exponent, the powers of "i" are: i, -1, -i, 1, i, etc... and this pattern repeats every 4 terms (notice how the 5th term is identical to the first term).


This pattern can be generalized to the following:

For (where "k" is a whole number),


If the remainder of is 0, then .


If the remainder of is 1, then .


If the remainder of is 2, then .


If the remainder of is 3, then .


So out at the 100th term, since remainder 0, this means that

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... Start with the given expression.



... Multiply the outer exponent by EVERY exponent in the parenthesis.


... Multiply


... Reduce


... Simplify


... Convert to radical notation.


... Take the cube root of 8 to get 2


... Square 2 to get 4


... Multiply



So

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Start with the given equation.


Multiply EVERY term by the LCD to clear any fractions.


Multiply.


Combine like terms on the left side.


Subtract from both sides.


Combine like terms on the left side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is

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Please don't type in all uppercase. It's hard to read that way.



Start with the given expression.


Factor to get .


Factor to get .


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where or

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Please don't type in all uppercase. It's hard to read that way.



Start with the given expression.


Factor to get .


Factor to get .


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where or

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Let's start off with some simple cases:

Case 1: If we have only one y, then we'll simply have


Case 2: If we add up two y's, then


Case 3: If we add up three y's, then


Case 4: If we add up four y's, then


etc...


So all we're doing is placing the number of y's that we're adding in front of the y's.


So if we add up 5 y's, then we'll get


If we add up 22 y's, then we'll get .


So if we add up "x" y's, then we'll get which is the answer.


Note: the reason why this works is because multiplication is really a shortcut for repeated addition.

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Start with the given inequality.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate . note: Remember, the inequality sign flips when we divide both sides by a negative number.


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is

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Start with the given equation


Combine the logs using the identity


Use the property ----> to rewrite the equation.


Multiply both sides by "x"


Divide both sides by


So the solution is

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# 1

Remember, probability is simply a formal way of representing the chance of an event occurring. Probability is most often represented as a fraction where the numerator is the number of ways that given event can occur over the denominator which represents the total number of possible outcomes.

In other words, this fraction looks like




where the notation "P(Event Occurring)" stands for "the probability of the event occurring"


So whenever you hear something like "the chances of that happening are one out of two", they mean that the event can occur one way out of two possible ways total. So they might be describing a coin toss where the chances of heads is "one out of two" since there's only one way to land on heads out of the two possible sides of the coin.


a)

Let's say that we have a deck of 52 cards and I want to know what the chances are of drawing a blue card. Since there are NO blue cards (there are only red and black cards), this means that the numerator is equal to zero while the denominator is still equal to 52. So this means that the probability of drawing a blue card is:



So the probability of choosing a blue card is 0 which means that there's a 0% chance of choosing a blue card.

----------------------

b)
Let's again say that we have a deck of 52 cards, but this time we want to know the chances of drawing a card (this is trivial, but important). Since there are 52 cards, and there are 52 ways to draw one card, this means that



So the probability of drawing a card is 1. In other words, no matter which card you choose, you're still selecting a card. So there is a 100% certainty that you'll select a card.


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# 2

The outcomes of an experiment are the results after running the simulation. Whatever experiment you run, the outcomes are the OBSERVED occurrences of an event.

For instance, let's say that you're flipping a coin and you observe the coin lands on: heads, tails, heads, heads, heads, and tails.

The outcomes of the experiment are the results of heads, tails, heads, heads, heads, and tails where the experiment is flipping a coin.

Note: an experiment in this sense is simply a simulation of an event (eg flipping a coin, rolling a die, drawing a card)

-----------------------------------------

Here's another example. Now you're rolling a single die and you roll the following: 1, 4, 2, 5, 1, 6, 2, and 3

The numbers 1, 4, 2, 5, 1, 6, 2, and 3 are all outcomes of the experiment of rolling a die.


Note: A more sophisticated experiment would involve more factors (and not be as simple as rolling a die or flipping a coin).


-----------------------------------------

When we say that the outcomes are equally likely, we're simply saying that the probability of the outcomes are equal.

Back in the coin flipping example, the chances of landing on heads is (since there's one way to get heads out of 2 possible outcomes). Likewise, the chances of getting tails is also (there's one way to get tails out of 2 possible outcomes).

Since , this means that the chances of getting either one is equally likely. So this means that the outcomes are equally likely.


In the die rolling example, the chances of rolling any number from 1 to 6 are (since for any given number, there's only one way to get that number out of the 6 numbers on the die). Since the chances of getting any number on the die is , these outcomes are also equally likely.

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From we can see that , , and


Start with the discriminant formula.


Plug in , , and


Square to get


Multiply to get


Subtract from to get


Since the discriminant is less than zero, this means that there are two complex solutions.


In other words, there are no real solutions.

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Start with the given expression.


Rewrite the logs using the identity


Combine the logs using the identity


So where every variable is positive.

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Since the times are the same, the time is simply "t" for both equations.


So we have the equations







Start with the first equation.


Divide both sides by .


Rearrange the equation


------------------------


Move onto the second equation


Plug in


Multiply both sides by .


Rearrange the terms.


Distribute.


Subtract from both sides.


Add to both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So the speed of the wind is 20 mph.

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Since we want to factor to , this means that we want


Start with the given equation.


FOIL


Combine like terms.


Distribute


Subtract from both sides.


Divide both sides by -1 to isolate "k".


So the solution is which means that factors to

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First, perform synthetic division on the polynomial using the test zero . Note: you can easily solve for "z" and find that one solution is

i |  1  0  0   i
  |     i  -1  -i
  -----------
     1  i  -1   0

Since the first three numbers in the third row are 1, i, and -1, this means that




Now simply use the quadratic formula to solve for "z":


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply


Combine like terms.


So the other two solutions are or


===========================================================

Answer:

So the 3 solutions are , , or

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I'm not sure how you got what you have, but I think you're confusing the area and perimeter formulas.

Since "A rectangle is 7 times as long as it is wide", this means that . Also, because "Its area is 1400cm^2", we know that


Recall, the area of a rectangle is


Start with the area of a rectangle formula


Plug in and


Multiply


Divide both sides by 7


Reduce


Rearrange the equation


Take the square root of both sides. Note: we're only considering the positive square root.


Factor 200 to get 100*2


Break up the square root.


Take the square root of 100 to get 10


Go back to the first equation


Plug in


Multiply



So the length is cm and the width is cm

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Start with the given equation


Square both sides


Square 5 to get 25


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the solution is

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Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is -2 and the run is 3. This means that to go from point to point, we can go down 2 and over 3



So starting at , go down 2 units


and to the right 3 units to get to the next point



Now draw a line through these points to graph



So this is the graph of through the points and

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Start with the given equation.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the answers are or

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Since the equation has the x- intercepts (2,0) and (-7,0), this means that the roots are or


or Start with the given roots.


or Get everything to the left side.


Combine the equations using the zero product property. Also, insert the variable "a" (this will be used to make the graph have a y-intercept of (0,5))


FOIL



So the general equation with the x- intercepts (2,0) and (-7,0) is:



Start with the given equation.


Distribute


Plug in and (from the point (0,5))


Multiply and simplify


Divide both sides by -14.


Reduce and rearrange the equation.



Go back to the previous equation


Plug in


Multiply and simplify



So the quadratic equation is


Here's a graph to visually confirm the answer:

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Note: because this is a quadratic and there are two terms that involve "x", you can't isolate "x" using linear techniques.


Start with the given equation.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the answers are or

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Start with the graph of





Now shift the graph over 2 units to the right to graph




Graph of (red) and (green)




Finally, shift the previous graph up 3 units to get the graph of




Graph of (red), (green), and (blue)


So the graph of is

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Start with the volume of a rectangular prism formula.


Plug in the volume of and the length and height of and . We'll use "x" for the unknown width.


Multiply


Divide both sides by 20 to isolate "x"


Reduce


So the solution is which means that the length is 6 meters.