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Start with the given equation.


Combine the logs using the identity


Rewrite the equation using the property: ====>


Raise 3 to the 4th power to get 81


FOIL


Subtract 81 from both sides.


Combine like terms.


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


Rewrite as


Add to to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


Break up the fraction.


Reduce.


or Break up the expression.


So the possible answers are or



However, since you cannot take the log of a negative number, this rules out


So the only solution is

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Start with the given equation.


Multiply every term on both sides by the LCD . Doing this will eliminate all of the fractions.


Multiply and simplify


Distribute


Get everything to the left side


Combine like terms.


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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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


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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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 c


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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# 1

The probability that the next call will be in person is simply the number of calls answered in person over the number of calls total. So



So the chances of a call answered in person are which in decimal form is 0.6 which gives a 60% chance


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# 2

Take note that there are 4 numbers that 5 on a single die namely: 1, 2, 3, and 4




So the relative frequency is which is 66.67%. So this means that the answer is A)

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# 3

First, we need to find the total number of marbles: 4 yellow + 12 green + 11 black = 27 total

So there are 27 marbles in total.



To find the chances of NOT yellow, we can do it two ways:

Method # 1:
Simply add up the number of marbles that are NOT yellow. So add up the 12 green and 11 black to get 23 marbles that are NOT yellow.

Now divide it by the total 27 to get:




So the chances of selecting a non yellow marble is which is 0.8519 in decimal form and gives a 85.19% chance of occurring

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

Method # 2:


If we pick a marble, it is either yellow or NOT yellow (but not both). There are no other choices.

So this means that if we find the chances of picking a yellow marble, we can subtract this from 1 to find the chances of getting a non yellow marble.

So....




So the chances of selecting a non yellow marble is again which is 0.8519 in decimal form and gives a 85.19% chance of occurring

So we get the same answer using either method. This means that you can use whatever you find easier (to compute or understand)


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# 4

If we look through a deck of cards, we'll find that there are 4 jacks (one of each suit)

So



So there is a chance of picking a jack. This in decimal form is giving a 7.69% chance.


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# 5

Since the odds against Thunderbolt winning are 11:3, this means that the corresponding odds are

Winning: 3
Losing: 11
giving a total of 11+3=14

So ...




This means that the probability of winning is giving a decimal value of 0.2143 which means that there's a 21.43% of Thunderbolt winning.

So the answer is B)

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# 6

a)

First, list out the individual probabilities for every possible scenario:

P($300 Prize) = 1/500
P($200 Prize) = 1/500
P($50 Prize) = 3/500
P(Losing) = 996/500 = 249/125


Now simply multiply the probabilities listed above with their corresponding winning value and then add them all up like so:


Expected Value = P($300 Prize)*(Value) + P($200 Prize)*(Value) + P($50 Prize)*(Value) + P(Losing)*(Value)


Expected Value = (1/500)*(300) + (1/500)*(200) + (3/500)*(50) + (496/500)*(0)


Expected Value = 300/500 + 200/500 + 150/500 + 0


Expected Value = 3/5 + 2/5 + 3/10


Expected Value = 6/10 + 4/10 + 3/10


Expected Value = (6 + 4 + 3)/10


Expected Value = 13/10


Expected Value = 1.30


So he expects to earn $1.30 on average (without factoring in the cost)


However, we need to subtract the cost of playing a single game to get

1.30 - 2 = -0.70

So he expects to lose 70 cents

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

b)

Since the expected winnings is $1.30, this means that the cost must equal the expected winnings to make the expected value equal to zero (since 1.30-1.30=0)

So the fair price for the ticket is $1.30

==================================================================================================
# 7

Let
r = red ball
b = blue ball
g = green ball
y = yellow ball


Let's list out ALL the possible points in the sample space

{r,b}
{r,g}
{r,y}
{b,r}
{b,g}
{b,y}
{g,r}
{g,b}
{g,y}
{y,r}
{y,b}
{y,g}


So there are 12 different possible outcomes which means that there are 12 points in the sample space.

Take note that first the first ball there are 4 choices and then once we select a ball (and don't put it back) we'll have 3 choices. Multiply these values together to get

4 * 3 = 12

So we get the same answer.


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# 8

We could list out all the possible license plates, but that would take too long and there's a very high chance that we'll repeat values and/or forget to list some elements.


Since the first slot must be a vowel, this means that there are 5 choices for the first position. For the second slot, there are 25 choices (since repetition of letters is not allowed). Similarly, there are 24 choices for the third slot.

For the digits, there are 10 choices (for numbers 0 - 9). Since repetition is allowed, there are 10 choices for the last digit also.

Now simply multiply all of the choices to get

5 * 25 * 24 * 10 * 10 = 125 * 240 * 10 = 300,000

So there are 300,000 different license plates.

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# 9

Simply use the counting principle to find the number of possible outfits:


Number of Pants * Number of Shirts * Number of Ties = 7 * 9 * 4 = 63 * 4 = 252


So there are 252 different outfits

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# 10

a)

Number of Genders * Number of Colors * Number of Seat Choices = 2 * 3 * 3 = 6 * 3 = 18

So there are 18 different bike configurations.

b)

Here's the tree:



c)

Here are ALL of the possible configurations which form the sample space:

{male, red, soft}
{male, red, medium}
{male, red, hard}
{male, white, soft}
{male, white, medium}
{male, white, hard}
{male, blue, soft}
{male, blue, medium}
{male, blue, hard}
{female, red, soft}
{female, red, medium}
{female, red, hard}
{female, white, soft}
{female, white, medium}
{female, white, hard}
{female, blue, soft}
{female, blue, medium}
{female, blue, hard}

Note: an example point {female, red, soft} means that the bike has a female frame, is red, and has a soft seat.

==================================================================================================
# 11

Let's use the table to answer these questions:

a)


So the chances are giving a decimal value of 0.4646 or a 46.46% chance
b)



So the chances are giving a decimal value of 0.4706 or a 47.06% chance
c)



So the probability is giving a decimal value of 0.5259 or a 52.59% chance


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# 12

For the first course, there are 6 choices. For the second there are 5 choices. For the third, there are 4 etc..


So multiplying all of this together, we get:





This means that there are 720 different ways for 6 instructors to teach six sections of a course in mathematics.


Note: the factorial button (!) is used as a shortcut. So


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# 13

a)

For first place, there are 7 choices. For second place, there are 6 choices (since it's not possible to place in two different positions). For third place, there are 5 choices, etc.. all the way down to 1.

So multiplying all of this together, we get




This means that there are 5,040 different arrangements.


b)

For first place, there are 7 choices. For second place, there are 6 choices and finally for third place there are 5 choices. Multiply these values together to get:

7*6*5 = 42*5 = 210

So there are 210 different ways to place in first, second and third place



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# 14

We could use the counting principle as we've done for earlier problems, but we'll have overlap and the sample space is far too large. So let's do it this way:


In this case, order does NOT matter since the candidates have no rank over one another (ie one isn't president or secretary).


Since order does not matter, we must use the combination formula:


Start with the combination formula.


Plug in and


Subtract to get 9


Expand 15!


Expand 9!


Cancel


Simplify


Expand 6!


Multiply 15*14*13*12*11*10 to get 3,603,600


Multiply 6*5*4*3*2*1 to get 720


Reduce.


So 15 choose 6 (where order doesn't matter) yields 5,005 unique combinations


This means that there are 5,005 different ways to select a group of 6 college candidates from a group of 15 applicants for an interview (where the order of the candidates doesn't matter).

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I'm assuming that the equation is ????



Start with the given equation.


Multiply EVERY term by the LCD to clear any fractions.


Multiply.


Combine like terms on the left side.


Add to both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is which approximates to .

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First, let's convert into an improper fraction





So the mixed fraction converts to the improper fraction


This means that



then becomes






Start with the given equation.


Multiply


Multiply both sides by 2


Multiply.


Divide both sides by to isolate .


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

Answer:

So the answer is which approximates to .

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# 1



Looking at we can see that the first term is and the last term is where the coefficients are 1 and 14 respectively.

Now multiply the first coefficient 1 and the last coefficient 14 to get 14. Now what two numbers multiply to 14 and add to the middle coefficient -9? Let's list all of the factors of 14:



Factors of 14:
1,2,7,14

-1,-2,-7,-14 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 14
1*14
2*7
(-1)*(-14)
(-2)*(-7)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9

First Number	Second Number	Sum
1	14	1+14=15
2	7	2+7=9
-1	-14	-1+(-14)=-15
-2	-7	-2+(-7)=-9

From this list we can see that -2 and -7 add up to -9 and multiply to 14


Now looking at the expression , replace with (notice combines 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 factors to

---

# 2



Looking at we can see that the first term is and the last term is where the coefficients are 6 and -4 respectively.

Now multiply the first coefficient 6 and the last coefficient -4 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient -5? Let's list all of the factors of -24:



Factors of -24:
1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -24
(1)*(-24)
(2)*(-12)
(3)*(-8)
(4)*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)

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


Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5

First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

From this list we can see that 3 and -8 add up to -5 and multiply to -24


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 factors to

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Start with the given expression.


Expand. Remember something like .


Now let's FOIL the expression.


Remember, when you FOIL an expression, you follow this procedure:


Multiply the First terms:.


Multiply the Outer terms:.


Multiply the Inner terms:.


Multiply the Last terms:.


---------------------------------------------------
So we have the terms: , , , and


Now add every term listed above to make a single expression.


Now combine like terms.


So FOILs to .


In other words, .

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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.


Reduce.


So .


So the expression is now in standard form where and

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First, simplify the numerator by adding the exponents:


So


Start with the given expression.


Subtract the exponents


Subtract


Raise 8 to the zeroth power to get 1


So which means that

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"I want to spend more than 6,000 " translates to and "less than 10,000" translates to


Now combine the inequalities to form

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# 1


Start with the given expression


Factor 54 into . Note: 9 is the largest perfect square factor of 54


Factor into


Break up the square root.


Take the square root of 9 to get 3


Take the square root of to get "x"


Multiply


So where

---

# 2


Start with the given expression


Simplify to get


Simplify to get


Multiply


Factor out the GCF


Combine like terms.



So where every variable is non negative.

---

# 3

Start with the given expression


Multiply both the numerator and denominator by


Multiply to get


Multiply to get


Reduce


So

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Start with the given equation.


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.




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 4 and the run is 1. This means that to go from point to point, we can go up 4 and over 1



So starting at , go up 4 units


and to the right 1 unit 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.


Square both sides


Square the square root of the left side to eliminate it.


Square 8 to get 64


Multiply EVERY term by the LCD to clear the fractions.


Distribute and multiply.


Subtract from both sides.


Combine like terms on the right side.


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

Answer:

So the answer is

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population growth rates and compounding interest is an example of exponential growth

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Take note that the first term is 14 and we're adding 3 to each term to get the next term. So this tells us that and


So the general formula (which is an arithmetic sequence) becomes

where "n" starts at 0


Note: if you want to start at n=1, then simply subtract 3 from 14 (to scale back the terms) to get where "n" starts at 1.

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Start with the given equation


Rewrite 6 as


Combine the roots.


Multiply


Square both sides. Take note that


So all you need to do is find two numbers that multiply to 180. I'll let you do that.

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Start with the given equation.


Combine like terms on the left side.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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# 1

a)

... Start with the given probability


... Plug in the given values.


... Reduce.


... Use a calculator (or long division) to find the approximate answer.


So the chances of a tulip blooming are 0.8378 which is 83.78% (multiply by 100 to get a percentage).

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

b)


If we multiply the number planted by the probability that a flower will bloom, we will get an approximate number that will bloom. This is because the probability allows us to estimate a population.

So let x = number originally planted and y = number of blooming flowers wanted.


So the equation is


Start with the given equation


Plug in (the probability we found earlier) and (the number of tulips wanted)


Divide both sides by 0.8378


Divide


Rearrange the equation.


Round up to the nearest whole number (this guarantees that we pass our target).


So if the gardener plants 114 or more bulbs, then at least 95 bulbs should bloom.


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# 2

Notation note: The notation n(...) means "the number of ...". So n(assualts) is read as "the number of assualts" and n(violent crimes) is read as "the number of violent crimes".


To find the chances of an assualt, simply divide the number of recorded assualts (17) by the total number violent crimes (50) like so:



So there is a 34% chance that the next violent crime will be an assualt.

Note: simply multiply the decimal value 0.34 by 100 to get 34%


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# 3


In this problem, the population of Pennsylvannia is extra information. In other words, you don't use this info to find the answer (and it's there to confuse you).


Number of people who do NOT live in California = Number people in US - Number of people who live in California

Number of people who do NOT live in California = 293,655,404 - 35,893,799 = 257,761,605
So
Number of people who do NOT live in California = 257,761,605

"What is the probability that a randomly selected US resident did not live in California? "

So the probability of selecting a US resident that does NOT live in California is...

P(NOT California) = # who do NOT live in California/# total
P(NOT California) = 257,761,605/293,655,404
P(NOT California) = 0.877769

So the percentage is 87.7769% which rounds to 87.777% (to the nearest thousandth of a percent).


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# 4

When they report the odds of 9:4, this means that the chances of losing is and the chances of winning are (note: which is the total). These odds form the ratio





and if we plug the given values in, we get the ratio:


Start with the given ratio


Multiply both sides by 8


Multiply


Simplify


So our answer is which means that Sally wins $18 (which makes the answer choice C)

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# 5

First, list out the probabilities for each possible event:

P(grand prize) = 1/1000
P(consolation prize) = 2/1000 = 1/500
P(losing) = 997/1000


Note: since there are 3 prize positions, there are 1000-3=997 losing positions.


Now simply multiply the probabilities with their corresponding winning value and then add up those products. In other words....

Expected Value = P(Grand Prize)*Value of Grand Prize + P(Consolation Prize)*Value of Consolation Prize + P(losing)*Value of losing


Now plug in the given values and evaluate:


Expected Value = (1/1000)*(100) + (1/500)*(75) + (997/1000)*(0)

Expected Value = 1/10 + 3/20 + 0

Expected Value = 5/20

Expected Value = 1/4

Expected Value = 0.25


So he expects to make $0.25. In other words, on an average try, he'll make 25 cents.

However, he spent $3 to play the game so this means that we need to subtract the cost to play from the expected winnings to get:

Expected Average Winnings - Cost to Play = $0.25 - $3 = -$2.75

So he expects to lose $2.75 overall which means that the answer is A).

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# 6

To find the expected value, simply perform a weighted average:


Weighted Average:



where

S = Number of Suspense Movies
s = Number of Suspense Movie Viewers
C = Number of Comedy Movies
c = Number of Comedy Movie Viewers
D = Number of Drama Movies
d = Number of Drama Movie Viewers

So plug in the given values and evaluate:













So the expected number of viewers is 5,025 which means that the answer is D)

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# 7


Take note that the even and odd digits are
Even: 0, 2, 4, 6, 8
Odd: 1, 3, 5, 7, 9

So there are 5 even and 5 odd numbers.

Also, the number of values from 0 to 9 is 10.

So to find the number of combinations, simply multiply the number of choices for each digit. So multiply the number of choices for the first digit (5) by the number of choices for the second digit (10) by the number of choices for the third digit (5):

5*10*5=50*5=250

So 250 different numbers can be formed based on the given criteria.

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

Side Note (ignore if you understand method given above): if you aren't sure about the multiplication, try using smaller numbers. So let's say we can only use the numbers 1, 2, and 4. How many 2 digit numbers can we form (with repeated values)?

Well, we can list them out:
11, 12, 14,
21, 22, 24,
41, 42, 44

From this list, we can see that there are 9 possible numbers. Take note that we have 3 choices for each digit. So multiplying 3*3 gets us 9 which is the original answer. Also, you can draw a tree to visually reinforce the idea.

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# 8

Whenever we want to find the probability of event A OR event B happening, we simply add the two probabilities like so:




... Start with the given probability


... Expand


... Plug in the given values.


... Add the fractions.


... Add


... Reduce


So the chances are which is 0.8 in decimal form which gives an 80% chance.


=====================================================================================================
# 9

a)



Using the tree, we get the sample space:

{b,b,b,b}
{b,b,b,g}
{b,b,g,b}
{b,b,g,g}
{b,g,b,b}
{b,g,b,g}
{b,g,g,b}
{b,g,g,g}

{g,b,b,b}
{g,b,b,g}
{g,b,g,b}
{g,b,g,g}
{g,g,b,b}
{g,g,b,g}
{g,g,g,b}
{g,g,g,g}

So for instance {b,g,g,b} means that the couple had a boy, girl, girl, and then a boy (in that order).

Note: recall, the sample space is the set of ALL possible outcomes.
-------------------------------

b)
Since we want to know the chances of the couple having "at least three boys", this means they want to know the chances of having 3 boys OR 4 boys (since at least means that figure or more).


Looking back at the list of all possible outcomes (ie the sample space) from part a), we see that we have the combinations for 3 boys:
{b,b,b,g}, {b,b,g,b}, {b,g,b,b}, and {g,b,b,b}

So there are 4 cases where the couple would have 3 boys.


So

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

Also, since there is only ONE way to have 4 boys (of a total of 4 children), this means that






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


Now simply add the two probabilities to find the chances of either one occurring:



















So the probability of the couple having AT LEAST 3 boys is which is 0.3125 in decimal form which gives a 31.25% chance

=====================================================================================================
# 10

a)

Since there are 12 face cards (there are 4 groups of jack, queen, and king cards) and there are 52 total cards, this means that there are cards that are NOT face cards.

Since we know that the given card is NOT a face card, this means we can focus solely on the 40 non face cards. So the number of the sample space is 40.

Also, since there are 4 cards labeled "7", this means that the number of elements in the event space is 4.

So




So the chances of drawing a 7 given that the card is NOT a face card is which is 0.1 or 10%


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

b)

Since spade cards are black and the given card is red, this means that the probability of choosing a spade given the card is red is 0%. In other words, it is IMPOSSIBLE to choose a spade card given the card is red.


More formally, we can say that


Number of spades that are red = 0
Number of Red Cards = 26


P(spade | red) = 0/26 = 0

So either way, we get the same answer of 0

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Start with the given equation.


Multiply EVERY term (that's outside the parenthesis), by 100 to make every number a whole number.


Distribute.


Combine like terms on the left side.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is

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Start with the given expression


Simplify to get .


Simplify to get .


Simplify to get .


Multiply 2 and 4 to get 8.

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


Factor out the GCF


Combine like terms.


So simplifies to .


In other words,

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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 t


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


or Break up the expression.


or Reduce


So the answers are or


which approximate to or

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Start with the given equation.


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.


So the equation is now in slope intercept form where the slope is and the y-intercept is

So the y-intercept is the point (0, -2)


So all we need to do is find the equation of the line through (0,-2) and (5,1)





First let's find the slope of the line through the points and


Note: is the first point and is the second point .


Start with the slope formula.


Plug in , , , and


Subtract from to get


Subtract from to get


So the slope of the line that goes through the points and is


Now let's use the point slope formula:


Start with the point slope formula


Plug in , , and


Rewrite as


Distribute


Multiply


Subtract 2 from both sides.


Combine like terms.


So the equation that goes through the points and is

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I'm assuming that order doesn't matter.


Since order does not matter, we must use the combination formula:


Start with the given formula



Plug in and



Subtract to get 4


Expand 9!



Expand 4!




Cancel



Simplify


Expand 5!




Multiply 9*8*7*6*5 to get 15,120



Multiply 5*4*3*2*1 to get 120



Now divide



So 9 choose 5 (where order doesn't matter) yields 126 unique combinations

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Hmm...45 seems like a very long distance. If you plot the points and use the grid, you'll quickly find that 45 is way too big...




Here's how you'd find the distance:


Start with the distance formula.


Plug in , , , and .


Subtract from to get .


Subtract from to get .


Square to get .


Square to get .


Add to to get .


Take the square root of to get .


So our answer is


So the distance between the two points (-2,1) and (6,7) is 10 units.

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Start with the given expression.


Factor into


Factor into


Factor into


Break up the square root using the identity .


Take the square root of to get .


Take the square root of to get .


Take the square root of to get .


Take the square root of to get .


Rearrange and multiply the terms.

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Answer:


So simplifies to


In other words, where every variable is non-negative.

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Are you sure that it's not a square root and the expression is ??? If so, then...


Take note that and . So this means that



Start with the given expression.


Replace with . Replace with (see above)


Use De Moivre's formula to expand.


Multiply


Evaluate the trig functions.


Simplify


So

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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


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 solutions are or

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You can't solve for "x" (this is not an equation), but you can simplify.


Start with the given expression.


Factor to get .


Factor to get .


Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


Simplify.


Distribute


So simplifies to .


In other words, where , , or

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Start with the given expression


Simplify to get .


Simplify to get .


Multiply 4 and 10 to get 40.

Multiply 8 and 5 to get 40.


Factor out the GCF


Combine like terms.


Multiply

So simplifies to .


In other words,