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



So starting at , go up 1 unit


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

Notice how the x-intercept is (-3,0) and the y-intercept is (0,3)

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Start with the given expression


Group like terms


Combine like terms

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Looking at we can see that the first term is and the last term is where the coefficients are 35 and -72 respectively.

Now multiply the first coefficient 35 and the last coefficient -72 to get -2520. Now what two numbers multiply to -2520 and add to the middle coefficient -23? Let's list all of the factors of -2520:



Factors of -2520:
1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,36,40,42,45,56,60,63,70,72,84,90,105,120,126,140,168,180,210,252,280,315,360,420,504,630,840,1260,2520

-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,-12,-14,-15,-18,-20,-21,-24,-28,-30,-35,-36,-40,-42,-45,-56,-60,-63,-70,-72,-84,-90,-105,-120,-126,-140,-168,-180,-210,-252,-280,-315,-360,-420,-504,-630,-840,-1260,-2520 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -2520
(1)*(-2520)
(2)*(-1260)
(3)*(-840)
(4)*(-630)
(5)*(-504)
(6)*(-420)
(7)*(-360)
(8)*(-315)
(9)*(-280)
(10)*(-252)
(12)*(-210)
(14)*(-180)
(15)*(-168)
(18)*(-140)
(20)*(-126)
(21)*(-120)
(24)*(-105)
(28)*(-90)
(30)*(-84)
(35)*(-72)
(36)*(-70)
(40)*(-63)
(42)*(-60)
(45)*(-56)
(-1)*(2520)
(-2)*(1260)
(-3)*(840)
(-4)*(630)
(-5)*(504)
(-6)*(420)
(-7)*(360)
(-8)*(315)
(-9)*(280)
(-10)*(252)
(-12)*(210)
(-14)*(180)
(-15)*(168)
(-18)*(140)
(-20)*(126)
(-21)*(120)
(-24)*(105)
(-28)*(90)
(-30)*(84)
(-35)*(72)
(-36)*(70)
(-40)*(63)
(-42)*(60)
(-45)*(56)

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


Now which of these pairs add to -23? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -23

First Number	Second Number	Sum
1	-2520	1+(-2520)=-2519
2	-1260	2+(-1260)=-1258
3	-840	3+(-840)=-837
4	-630	4+(-630)=-626
5	-504	5+(-504)=-499
6	-420	6+(-420)=-414
7	-360	7+(-360)=-353
8	-315	8+(-315)=-307
9	-280	9+(-280)=-271
10	-252	10+(-252)=-242
12	-210	12+(-210)=-198
14	-180	14+(-180)=-166
15	-168	15+(-168)=-153
18	-140	18+(-140)=-122
20	-126	20+(-126)=-106
21	-120	21+(-120)=-99
24	-105	24+(-105)=-81
28	-90	28+(-90)=-62
30	-84	30+(-84)=-54
35	-72	35+(-72)=-37
36	-70	36+(-70)=-34
40	-63	40+(-63)=-23
42	-60	42+(-60)=-18
45	-56	45+(-56)=-11
-1	2520	-1+2520=2519
-2	1260	-2+1260=1258
-3	840	-3+840=837
-4	630	-4+630=626
-5	504	-5+504=499
-6	420	-6+420=414
-7	360	-7+360=353
-8	315	-8+315=307
-9	280	-9+280=271
-10	252	-10+252=242
-12	210	-12+210=198
-14	180	-14+180=166
-15	168	-15+168=153
-18	140	-18+140=122
-20	126	-20+126=106
-21	120	-21+120=99
-24	105	-24+105=81
-28	90	-28+90=62
-30	84	-30+84=54
-35	72	-35+72=37
-36	70	-36+70=34
-40	63	-40+63=23
-42	60	-42+60=18
-45	56	-45+56=11

From this list we can see that 40 and -63 add up to -23 and multiply to -2520


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


So one of the factors of is D)

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Start with the given system





Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.


Distribute


Combine like terms on the left side


Add 21 to both sides


Combine like terms on the right side


Divide both sides by 8 to isolate y



Divide




Now that we know that , we can plug this into to find



Substitute for each


Simplify


So our answer is and which also looks like



Notice if we graph the two equations, we can see that their intersection is at . So this verifies our answer.


Graph of (red) and (green)

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Start with the given equation

Subtract 12 from both sides

Factor the left side


or Set each factor equal to zero


or Solve for t in each case



---------------------------------
Answer:

So the solutions are


or
So our answer is C

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Start with the given expression


Let . So our expression becomes

Plug in





Looking at we can see that the first term is and the last term is where the coefficients are 6 and -35 respectively.

Now multiply the first coefficient 6 and the last coefficient -35 to get -210. Now what two numbers multiply to -210 and add to the middle coefficient -1? Let's list all of the factors of -210:



Factors of -210:
1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210

-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -210
(1)*(-210)
(2)*(-105)
(3)*(-70)
(5)*(-42)
(6)*(-35)
(7)*(-30)
(10)*(-21)
(14)*(-15)
(-1)*(210)
(-2)*(105)
(-3)*(70)
(-5)*(42)
(-6)*(35)
(-7)*(30)
(-10)*(21)
(-14)*(15)

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


Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1

First Number	Second Number	Sum
1	-210	1+(-210)=-209
2	-105	2+(-105)=-103
3	-70	3+(-70)=-67
5	-42	5+(-42)=-37
6	-35	6+(-35)=-29
7	-30	7+(-30)=-23
10	-21	10+(-21)=-11
14	-15	14+(-15)=-1
-1	210	-1+210=209
-2	105	-2+105=103
-3	70	-3+70=67
-5	42	-5+42=37
-6	35	-6+35=29
-7	30	-7+30=23
-10	21	-10+21=11
-14	15	-14+15=1

From this list we can see that 14 and -15 add up to -1 and multiply to -210


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

Now replace z with . Remember, we let


Distribute


Combine like terms



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

Answer:

So factors to

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Start with the given equation


Subtract 3 from both sides.



Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:
or

or Now solve for x in each case


So our solutions are

or

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Start with the given expression


Factor out the GCF


Factor to get by using the difference of squares



So one of the factors of are , or

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Start with the given equation

Let's find the x-intercept

To find the x-intercept, let y=0 and solve for x:
Plug in

Simplify

Divide both sides by -5


Reduce



So the x-intercept is (note: the x-intercept will always have a y-coordinate equal to zero)



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

Start with the given equation

Now let's find the y-intercept

To find the y-intercept, let x=0 and solve for y:
Plug in

Simplify

Divide both sides by 3



Reduce



So the y-intercept is (note: the y-intercept will always have a x-coordinate equal to zero)

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

So we have these intercepts:
x-intercept:

y-intercept:

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Start with the given expression

Factor to get

Cancel like terms

Simplify



--------------------------------------------------
Answer:

So simplifies to . In other words

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Using Descartes' Rule of Signs, we can find the possible number of positive roots (x-intercepts that are positive) and negative roots (x-intercepts that are negative)

First lets find the number of possible positive real roots:

For , simply count the sign changes

Here is the list of sign changes:

1. to (positive to negative)
2. to (negative to positive)

So there are 2 sign changes, this means there are a maximum of 2 positive roots

So there could be 2, or 0 positive real roots

---

Now to find the number of negative real roots, we need to find





Replace each x with


Simplify


Now let's count the sign changes for




Here is the list of sign changes:

1. to (negative to positive)
2. to (positive to negative)
3. to (negative to positive)

So there are 3 sign changes, this means there are a maximum of 3 negative roots

So there could be 3, or 1 negative real roots

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Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients


So let's list the factors of 12 (the last coefficient):



Now let's list the factors of 2 (the first coefficient):



Now let's divide each factor of the last coefficient by each factor of the first coefficient









Now simplify

These are all the distinct rational zeros of the function that could occur

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Start with the given equation


Subtract from both sides


Rearrange the equation


Divide both sides by


Break up the fraction


Reduce



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



So you have the correct answer.

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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 9 and the run is 5. This means that to go from point to point, we can go up 9 and over 5



So starting at , go up 9 units


and to the right 5 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 compound interest formula

Plug in p=500, r=0.05 (this is the decimal form of 5% interest), n=4, and t=10


Divide 0.05 by 4 to get 0.0125


Multiply the exponents 4 and 10 to get 40


Add 1 and 0.0125 to get 1.0125


Raise 1.0125 to the 40 th power to get 1.64361946348701


Multiply 500 and 1.64361946348701 to get 821.809731743505

So if you invest $500 at an interest rate of 5%, which is compounded 4 times a year for 10 years, the return is about $821.81 (which is rounded to the nearest cent)

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Start with the given equation



Take the square root of both sides




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



Add 5 to both sides to isolate n.



-----------------------------------
Answer:

So our solution is:



This means our solution breaks down to:

or

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Start with the given equation


Distribute



Combine like terms



Factor the left side



Now set each factor equal to zero:
or

or Now solve for x in each case



So our solutions are

or


which means that the x-intercepts are

(-3,0) and (0,0)



Notice if we graph we can see that the roots are and . So this visually verifies our answer.

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Remember the perimeter of a rectangle is



Since one side is formed from the side of the barn, this means that we can take out one length (or width, it doesn't matter) to get





Plug in the given perimeter 60 (since he only has 60 ft of fencing)

Subtract from both sides


Rearrange the equation



Now let's introduce another formula. The area of any rectangle is




Plug in


Rearrange the terms


Distribute


Rearrange the terms


From now on, let's think of as where y is the area and x is the width.




Now the equation is in the form of a quadratic which has a vertex that corresponds with the maximum area. So if we find the y-coordinate of the vertex, we can find the max area.





In order to find find the vertex, we first need to find the axis of symmetry (ie the x-coordinate of the vertex)
To find the axis of symmetry, use this formula:



From the equation we can see that a=-2 and b=60

Plug in b=60 and a=-2


Multiply 2 and -2 to get -4



Reduce


So the axis of symmetry is


So the x-coordinate of the vertex is . Lets plug this into the equation to find the y-coordinate of the vertex.


Lets evaluate

Start with the given polynomial


Plug in


Raise 15 to the second power to get 225


Multiply 2 by 225 to get 450


Multiply 60 by 15 to get 900


Now combine like terms


So the vertex is (15,450)

This shows us that the max area is then 450 square feet.



So with a width of 15 ft the fence will have a maximum area of 450 square feet



Now plug in


Multiply


Subtract


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

Answer:

So the dimensions of the garden are

width: 15, length: 30

Also, the max area of the garden is 450 square feet.

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To find the vertex, we first need to find the line of symmetry (ie the x-coordinate of the vertex)
To find the line of symmetry, use this formula:



From the equation we can see that a=-1 and b=2

Plug in b=2 and a=-1


Multiply 2 and -1 to get -2



Reduce


So the line of symmetry is


So the x-coordinate of the vertex is . Lets plug this into the equation to find the y-coordinate of the vertex.


Lets evaluate

Start with the given polynomial


Plug in


Raise 1 to the second power to get 1


Multiply 2 by 1 to get 2


Now combine like terms


So the vertex is (1,4)



-----------------------
Answer:

So the line of symmetry is and the vertex is (1,4)



If we graph, we can visually verify our answer

Graph of with the line of symmetry and the vertex (1,4)

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To find the vertex, we first need to find the line of symmetry (ie the x-coordinate of the vertex)
To find the line of symmetry, use this formula:



From the equation we can see that a=3 and b=-24

Plug in b=-24 and a=3


Negate -24 to get 24


Multiply 2 and 3 to get 6



Reduce


So the line of symmetry is


So the x-coordinate of the vertex is . Lets plug this into the equation to find the y-coordinate of the vertex.


Lets evaluate

Start with the given polynomial


Plug in


Raise 4 to the second power to get 16


Multiply 3 by 16 to get 48


Multiply 24 by 4 to get 96


Now combine like terms


So the vertex is (4,3)




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

Answer:


So the line of symmetry is and the vertex is (4,3)



If we graph, we can visually verify our answer

Graph of with the line of symmetry and the vertex (4,3)

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If we look at the page numbers, we'll notice that they are consecutive integers. So they follow the algebraic form , , , etc.



So if the product of two consecutive integers is 420, then we have the equation




Distribute


Subtract 420 from both sides



Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:
or

or Now solve for x in each case


So our possible solutions are

or



However, since a negative page number is not possible, this means that the only solution is


Now to find the next page number, simply add 1 to 20 to get





So the two page numbers are 20 and 21



Check:

To check this answer, simply multiply the two numbers:





Works.

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Start with the given expression

Factor to get


Cancel like terms


Simplify



So simplifies to


In other words,

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Start with the given equation



Subtract 15 from both sides


Subtract 6h from both sides


Divide both sides by -6 to isolate h



Reduce

--------------------------------------------------------------
Answer:
So our answer is (which is approximately in decimal form)

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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 x=measure of angle A

If we draw the triangle, it would look something like this:


To find angle A, we can use trigonometry to do so. We can use the tangent function to find the measure of angle A

Remember


So




Now take the inverse tangent of both sides


Take the inverse tangent of 1.8 to get 60.95



So the measure of angle A is about 60.95 degrees

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Start with the given equation


Distribute the terms on the left side


Subtract 16 from both sides


Divide both sides by 4 to isolate x

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I'll do the first two to get you started



# 1




Start with the given system of equations:





Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

Start with the first equation


Subtract from both sides


Rearrange the equation


Divide both sides by


Break up the fraction


Reduce



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

Since , we can now replace each in the second equation with to solve for



Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Multiply both sides by the LCM of 3. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



Distribute and multiply the LCM to each side



Combine like terms on the left side


Add 28 to both sides


Combine like terms on the right side


Divide both sides by -13 to isolate x



Divide





-----------------First Answer------------------------------


So the first part of our answer is:









Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms and reduce. (note: if you need help with fractions, check out this solver)



-----------------Second Answer------------------------------


So the second part of our answer is:









-----------------Summary------------------------------

So our answers are:

and

which form the point








Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of (red) and (green) and the intersection of the lines (blue circle).

---

Start with the given system of equations:





Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for n.




So let's isolate n in the first equation

Start with the first equation


Subtract from both sides


Rearrange the equation


Divide both sides by


Break up the fraction


Reduce



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

Since , we can now replace each in the second equation with to solve for



Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute to


Multiply


Combine like terms on the left side


Subtract 56 from both sides


Combine like terms on the right side


Divide both sides by 17 to isolate m



Divide





-----------------First Answer------------------------------


So the first part of our answer is:









Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms



-----------------Second Answer------------------------------


So the second part of our answer is:









-----------------Summary------------------------------

So our answers are:

and

which form the point (note: simply replace m with x and replace n with y)








Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of (red) and (green) and the intersection of the lines (blue circle).

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Start with the given system of equations:





Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

Start with the first equation


Subtract from both sides


Rearrange the equation


Divide both sides by


Break up the fraction


Reduce



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

Since , we can now replace each in the second equation with to solve for



Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute to


Multiply


Multiply both sides by the LCM of 3. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



Distribute and multiply the LCM to each side



Combine like terms on the left side


Subtract 16 from both sides


Combine like terms on the right side


Divide both sides by -2 to isolate x



Divide





-----------------First Answer------------------------------


So the first part of our answer is:









Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms and reduce. (note: if you need help with fractions, check out this solver)



-----------------Second Answer------------------------------


So the second part of our answer is:









-----------------Summary------------------------------

So our answers are:

and

which form the point








Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of (red) and (green) and the intersection of the lines (blue circle).

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Start with the given function


Plug in


Raise -1 to the 2nd power to get 1


Multiply -1 and 1 to get -1


Raise -1 to the 3rd power to get -1


Multiply 1 and -1 to get -1


Subtract -1 from -1 to get 0


Multiply 10 and -1 to get -10


Add 0 and -10 to get -10

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Start with the given system





Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.




Subtract 10 from both sides


Subtract 6x from both sides


Combine like terms on the left side


Combine like terms on the right side


Divide both sides by 3 to isolate x






Now that we know that , we can plug this into to find



Substitute for each


Simplify


So our answer is and which also looks like



Notice if we graph the two equations, we can see that their intersection is at . So this verifies our answer.


Graph of (red) and (green)