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I'll do the first two, which will hopefully help you with the rest of the problems. If not, then repost.


# 1



Start with the given expression


Factor out the GCF


Now let's focus on the inner expression




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



Looking at we can see that the first term is and the last term is where the coefficients are 15 and 8 respectively.

Now multiply the first coefficient 15 and the last coefficient 8 to get 120. Now what two numbers multiply to 120 and add to the middle coefficient 22? Let's list all of the factors of 120:



Factors of 120:
1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120

-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 120
1*120
2*60
3*40
4*30
5*24
6*20
8*15
10*12
(-1)*(-120)
(-2)*(-60)
(-3)*(-40)
(-4)*(-30)
(-5)*(-24)
(-6)*(-20)
(-8)*(-15)
(-10)*(-12)

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


Now which of these pairs add to 22? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 22

First Number	Second Number	Sum
1	120	1+120=121
2	60	2+60=62
3	40	3+40=43
4	30	4+30=34
5	24	5+24=29
6	20	6+20=26
8	15	8+15=23
10	12	10+12=22
-1	-120	-1+(-120)=-121
-2	-60	-2+(-60)=-62
-3	-40	-3+(-40)=-43
-4	-30	-4+(-30)=-34
-5	-24	-5+(-24)=-29
-6	-20	-6+(-20)=-26
-8	-15	-8+(-15)=-23
-10	-12	-10+(-12)=-22

From this list we can see that 10 and 12 add up to 22 and multiply to 120


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 )



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




So our expression goes from and factors further to


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

So completely factors to

---

# 2




Start with the given expression


Factor out the GCF


Now let's focus on the inner expression




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



Looking at we can see that the first term is and the last term is where the coefficients are 12 and 1 respectively.

Now multiply the first coefficient 12 and the last coefficient 1 to get 12. Now what two numbers multiply to 12 and add to the middle coefficient 7? Let's list all of the factors of 12:



Factors of 12:
1,2,3,4,6,12

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

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

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


Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7

First Number	Second Number	Sum
1	12	1+12=13
2	6	2+6=8
3	4	3+4=7
-1	-12	-1+(-12)=-13
-2	-6	-2+(-6)=-8
-3	-4	-3+(-4)=-7

From this list we can see that 3 and 4 add up to 7 and multiply to 12


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 )



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




So our expression goes from and factors further to


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

So completely factors to

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Let x=length of ladder


Since "the distance from the ground to the top of the ladder in one foot less than the length of the ladder.", this means that one leg of the triangle is ft. Also, we're given the other leg of 5 ft.


So this tells us that , and


Start with Pythagorean's Theorem


Plug in , and


Square 5 to get 25


FOIL


Subtract from both sides.


Combine like terms.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is


So this means that the length of the ladder is 13 ft while the height that the ladder reaches on the building is 12 ft.

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We can see that the equation has a slope and a y-intercept .


Now to find the slope of the perpendicular line, simply flip the slope to get . Now change the sign to get . So the perpendicular slope is .


Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope and the coordinates of the given point .


Start with the point slope formula


Plug in , , and


Rewrite as


Multiply both sides by 4.


Distribute


Subtract 12 from both sides.


Subtract "x" from both sides.


Combine like terms.


Rearrange the terms.


Multiply EVERY term by -1 to make the "x" coefficient positive.


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

Answer:

So the equation of the line that is perpendicular to and goes through the point (2,-3) in standard form is



Here's the graph of the two lines to verify the answer:


Graph of the original equation (red) and the perpendicular line (green) through the point .

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Start with the given equation.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.


We can see that the equation has a slope and a y-intercept .


Since parallel lines have equal slopes, this means that we know that the slope of the unknown parallel line is .
Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope and the coordinates of the given point .


Start with the point slope formula


Plug in , , and


Rewrite as


Rewrite as


Multiply both sides by 3.


Distribute


Add 5x to both sides.


Subtract 3 from both sides.


Combine and rearrange the terms.



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

Answer:

So the equation of the line that is parallel to and that goes through (-2,-1) is:


Also, the equation is in standard form where , , and

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Start with the given expression.


Combine the fractions. This is only possible if the denominators are equal (and they are)


Distribute


Combine like terms.


So where

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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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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 second equation

Start with the second equation


Subtract from both sides


Rearrange the equation


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

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 80 from both sides


Combine like terms on the right side


Divide both sides by 11 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



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


Divide both sides by -21 to isolate x



Reduce




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



Substitute for each


Multiply


So our answer is and which also looks like

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Start with the given expression.


Rewrite as .


Rewrite as .


Notice how we have a difference of squares where in this case and .


So let's use the difference of squares formula to factor the expression:


Start with the difference of squares formula.


Plug in and .


So this shows us that factors to .


In other words .

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Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
1,2,7,14
-1,-2,-7,-14


Note: list the negative of each factor. This will allow us to find all possible combinations.


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number	Second Number	Sum
1	-14	1+(-14)=-13
2	-7	2+(-7)=-5
-1	14	-1+14=13
-2	7	-2+7=5

From the table, we can see that the two numbers and add to (the middle coefficient).


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


Combine like terms. Or factor out the common term

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


Answer:


So factors to .


Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

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Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to .
1*(-30)
2*(-15)
3*(-10)
5*(-6)
(-1)*(30)
(-2)*(15)
(-3)*(10)
(-5)*(6)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number	Second Number	Sum
1	-30	1+(-30)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

From the table, we can see that the two numbers and add to (the middle coefficient).


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


Combine like terms. Or factor out the common term

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


Answer:


So factors to .


Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

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Start with the given expression


Factor the numerator


Factor the denominator


Highlight the common terms.


Cancel out the common terms.


Simplify


Distribute


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

Answer:


So simplifies to


In other words, where

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Start with the general slope-intercept equation


Plug in and


So the equation of the line is

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Start with the given expression


Factor out the GCF


Break down into


Highlight the common terms.


Cancel out the common terms.


Simplify (by removing the canceled terms)


Distribute


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

Answer:

So simplifies to


In other words, where

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Start with the second equation


Plug in . In other words, replace every "y" with 3x-4


Combine like terms


Add 4 to both sides


Divide both sides by 4 to isolate "x"


So the first part of the answer is


Go back to the first equation


Plug in


Multiply


Subtract

So the second part of the answer is

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

Answer:


So the solutions are and which form the ordered pair (3,5)


I'll let you do the check (simply plug in the two solutions and simplify)

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Notice how the line is of the form where "m" is the slope and "b" is the y-intercept.


So this simply means that and which tells us that the slope is 3 and the y-intercept is 4

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General circle equation:

Note: (h,k) is the center and "r" is the radius

So in our case, , and , since the center is (-2,4), and (since the radius is 3)


Start with the given equation


Plug in , , and


Square 3 to get 9


Rewrite as


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

Answer:

So the equation of the circle is

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Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition

Lets start with the given system of linear equations In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa). So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero. So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and -1 to some equal number, we could try to get them to the LCM. Since the LCM of 1 and -1 is -1, we need to multiply both sides of the top equation by -1 and multiply both sides of the bottom equation by -1 like this: Multiply the top equation (both sides) by -1 Multiply the bottom equation (both sides) by -1 So after multiplying we get this: Notice how -1 and 1 add to zero (ie ) Now add the equations together. In order to add 2 equations, group like terms and combine them Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether. So after adding and canceling out the x terms we're left with: Divide both sides by to solve for y Reduce Now plug this answer into the top equation to solve for x Plug in Multiply Subtract from both sides Combine the terms on the right side Multiply both sides by . This will cancel out on the left side. Multiply the terms on the right side So our answer is , which also looks like (, ) Notice if we graph the equations (if you need help with graphing, check out this solver) we get graph of (red) (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle). and we can see that the two equations intersect at (,). This verifies our answer.

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Note:

So


This means that




So


Now add the exponents and multiply:


So where

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Solved by pluggable solver: Finding the Equation of a Line

First lets find the slope through the points (,) and (,) Start with the slope formula (note: (,) is the first point (,) and (,) is the second point (,)) Plug in ,,, (these are the coordinates of given points) Subtract the terms in the numerator to get . Subtract the terms in the denominator to get Reduce So the slope is ------------------------------------------------ Now let's use the point-slope formula to find the equation of the line: ------Point-Slope Formula------ where is the slope, and (,) is one of the given points So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Distribute Multiply and to get Add to both sides to isolate y Combine like terms and to get ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line which goes through the points (,) and (,) is: The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver) Graph of through the points (,) and (,) Notice how the two points lie on the line. This graphically verifies our answer.

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I'll do the first one to get you started

d)



Start with the given system of equations:





In order to graph these equations, we must solve for y first.


Let's graph the first equation:


Start with the first equation.


Subtract from both sides.


Divide both sides by to isolate .


Rearrange the terms and simplify.


Now let's graph the equation:


Graph of .


Note: let me know if you need help graphing equations

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


Now let's graph the second equation :

Graph of .


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


Now let's graph the two equations together:


Graph of (red). Graph of (green)


From the graph, we can see that the two lines intersect at the point . So the solution to the system of equations is . This tells us that the system of equations is consistent and independent.

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Let's simplify

Error, solver not defined for name 'monomial_arithmetic'.

Error occurred executing solver 'monomial_arithmetic' .

So this means that


So the answer is





which means that the answer choice is A)

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Is the expression ??? If so, then...


Error, solver not defined for name 'monomial_arithmetic'.

Error occurred executing solver 'monomial_arithmetic' .

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Start with the given equation.


Multiply EVERY term by the LCD to clear out the fractions.


Multiply and simplify


FOIL


Distribute


Subtract y from both sides. Subtract 2 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 y


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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You CANNOT solve this (since there is no equals sign). So I'm assuming that you just want to simplify.



Start with the given expression.


Rewrite as


Multiply


So simplifies to

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In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula: .


Start with the given formula.


From , we can see that , , and .


Plug in and .


Multiply 2 and to get .


Divide.


So the x-coordinate of the vertex is . Note: this means that the axis of symmetry is also .


Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So the y-coordinate of the vertex is .


So the vertex is .


Now because (which is positive), this means that the parabola opens upward and that there is a minimum.

So the min is the same as the y-coordinate of the vertex. This means that the min is


Here's a graph to confirm our answers:


Graph of

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Start with the given equation.

Let (notice there are two terms that contain )

Plug in


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


Let's use the quadratic formula to solve for z


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 .


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 answers (in terms of z) are or


However, we need the solutions in terms of "x". Remember, we let , so...


or Go back to the solutions (in terms of "z")


or Plug in


or Add 1 to both sides (for each equation).


or Combine like terms.


or Divide both sides by 2 (for each equation) to isolate "x".


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

Answer:


So the solutions are or

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From we can see that , , and


Start with the discriminant formula.


Plug in , , and


Square to get


Multiply to get


Rewrite as


Add to to get


Since the discriminant is greater than zero, this means that there are two real solutions.

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Start with the given equation.


Distribute.


Combine like terms on the right 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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a)

Start with the distance rate time formula


Plug in d=800 and t=x


Divide both sides by x.


Rearrange the terms.


So the rational expression for his average speed is


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

b)

Start with the distance rate time formula


Plug in (since he traveled the same average speed) and t=6


Multiply



So the rational expression for his distance is

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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


Which approximates to



So the answer in interval notation is


Also, the answer in set-builder notation is


Here's the graph of the solution set on a number line:

Graph of the solution set where the open circle denotes the excluded value.