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Start with the given expression.


Distribute


Multiply


Combine like terms.


So

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Do you want to graph this? Please post full instructions.

If you want to graph, then...




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



So starting at , go up 1 unit


and to the right 4 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 system of equations:



Multiply the both sides of the first equation by 2.


Distribute and multiply.


Multiply the both sides of the second equation by -3.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms.


Simplify.


Divide both sides by to isolate .


Reduce.


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


Now go back to the first equation.


Plug in .


Multiply.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So our answer is and .


Which form the ordered pair .


This means that the system is consistent and independent.


Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer.


Graph of (red) and (green)

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I'll do the first three to get you started:


# 46


Start with the given expression


Factor out the GCF


So factors to

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

# 60


Looking at we can see that the first term is and the last term is where the coefficients are 10 and -2 respectively.

Now multiply the first coefficient 10 and the last coefficient -2 to get -20. Now what two numbers multiply to -20 and add to the middle coefficient 1? Let's list all of the factors of -20:



Factors of -20:
1,2,4,5,10,20

-1,-2,-4,-5,-10,-20 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -20
(1)*(-20)
(2)*(-10)
(4)*(-5)
(-1)*(20)
(-2)*(10)
(-4)*(5)

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	-20	1+(-20)=-19
2	-10	2+(-10)=-8
4	-5	4+(-5)=-1
-1	20	-1+20=19
-2	10	-2+10=8
-4	5	-4+5=1

From this list we can see that -4 and 5 add up to 1 and multiply to -20


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


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

# 80



Looking at we can see that the first term is and the last term is where the coefficients are 4 and 25 respectively.

Now multiply the first coefficient 4 and the last coefficient 25 to get 100. Now what two numbers multiply to 100 and add to the middle coefficient 20? Let's list all of the factors of 100:



Factors of 100:
1,2,4,5,10,20,25,50

-1,-2,-4,-5,-10,-20,-25,-50 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 100
1*100
2*50
4*25
5*20
10*10
(-1)*(-100)
(-2)*(-50)
(-4)*(-25)
(-5)*(-20)
(-10)*(-10)

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


Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20

First Number	Second Number	Sum
1	100	1+100=101
2	50	2+50=52
4	25	4+25=29
5	20	5+20=25
10	10	10+10=20
-1	-100	-1+(-100)=-101
-2	-50	-2+(-50)=-52
-4	-25	-4+(-25)=-29
-5	-20	-5+(-20)=-25
-10	-10	-10+(-10)=-20

From this list we can see that 10 and 10 add up to 20 and multiply to 100


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 )


note: is equivalent to since the term occurs twice. So also factors to



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



Answer:
So factors to

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Start with the given equation.


Distribute.


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

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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 :
1,5,7,35
-1,-5,-7,-35


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


These factors pair up and multiply to .
1*(-35)
5*(-7)
(-1)*(35)
(-5)*(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	-35	1+(-35)=-34
5	-7	5+(-7)=-2
-1	35	-1+35=34
-5	7	-5+7=2

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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Table of Contents:
Step 1: Finding the Vertex
Step 2: Finding two points to left of axis of symmetry
Step 3: Reflecting two points to get points right of axis of symmetry
Step 4: Plotting the Points (with table)
Step 5: Graphing the Parabola

In order to graph , we can follow the steps:


Step 1) Find the vertex (the vertex is the either the highest or lowest point on the graph). Also, the vertex is at the axis of symmetry of the parabola (ie it divides it in two).


Step 2) Once you have the vertex, find two points on the left side of the axis of symmetry (the line that vertically runs through the vertex).


Step 3) Reflect those two points over the axis of symmetry to get two more points on the right side of the axis of symmetry.


Step 4) Plot all of the points found (including the vertex).


Step 5) Draw a curve through all of the points to graph the parabola.


Let's go through these steps in detail


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Step 1)

Finding the vertex:

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 .


Combine like terms.


So the y-coordinate of the vertex is .


So the vertex is .


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


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Step 2)

Find two points to the left of the axis of symmetry:

Let's find the y value when


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Combine like terms.


So the first point to the left of the axis of symmetry is (-2,2)


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


Let's find the y value when


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Combine like terms.


So the second point to the left of the axis of symmetry is (-1,-1)


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


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Step 3)

Reflecting the two points over the axis of symmetry:

Now remember, the parabola is symmetrical about the axis of symmetry (which is )


This means the y-value for (which is one unit from the axis of symmetry) is equal to the y-value of (which is also one unit from the axis of symmetry). So when , which gives us the point (1,-1). So we essentially reflected the point (-1,-1) over to (1,-1).


Also, the y-value for (which is two units from the axis of symmetry) is equal to the y-value of (which is also two units from the axis of symmetry). So when , which gives us the point (2,2). So we essentially reflected the point (-2,2) over to (2,2).


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


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Step 4)

Plotting the points:

Now lets make a table of the values we have calculated:

x	y
-2	2
-1	-1
0	-2
1	-1
2	2

Now let's plot the points:



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


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Step 5)

Drawing a curve through all of the points:

Now draw a curve through all of the points to graph :


Graph of

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a)


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

So the discriminant is 21.
----------------------------------------------------

b)

From part a), we see that . This means that (ie the discriminant is positive)

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

So we can say that...

Type of Solution(s): Real
Number: 2 (these are distinct)


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

c)



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 .


or Break up the expression.


So the answers are or


which approximate to or


Notice how there are two real solutions. So this confirms our answer to part b)

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To graph ANY function, simply follow this basic routine:

1) Plug in any x value to find it's corresponding function value (or y value). This gives you an ordered pair (x,y)

2) Plot the points that you calculated from step 1

3) Draw a smooth connected through ALL of the points that you plotted in step 2



I'll do the first two to get you started. The other two follow the same basic outline.

# 1



In order to graph , we need to plot a few points.


Start with the given equation.


Plug in (note: you can start at any x-value).


Raise 5 to the -2nd power to get .


So when , . So we have the point (-2,0.04).


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


Start with the given equation.


Plug in .


Raise 5 to the -1st power to get .


So when , . So we have the point (-1,0.2).


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


Start with the given equation.


Plug in .


Raise 5 to the 0th power to get .


So when , . So we have the point (0,1).


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


Start with the given equation.


Plug in .


Raise 5 to the 1st power to get .


So when , . So we have the point (1,5).


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


Now let's make a table of the values we just found.

Table of Values:

x
y
-2
0.04
-1
0.2
0
1
1
5

Now let's plot the points:

Graph:

Now draw a curve through all of the points to graph :


Graph of



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

# 2



In order to graph , we need to plot a few points.


Start with the given equation.


Plug in (note: you can start at any x-value).


Add.


Raise 4 to the -2nd power to get .


So when , . So we have the point (-4,0.063).


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


Start with the given equation.


Plug in .


Add.


Raise 4 to the -1st power to get .


So when , . So we have the point (-3,0.25).


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


Start with the given equation.


Plug in .


Add.


Raise 4 to the 0th power to get .


So when , . So we have the point (-2,1).


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


Start with the given equation.


Plug in .


Add.


Raise 4 to the 1st power to get .


So when , . So we have the point (-1,4).


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


Now let's make a table of the values we just found.

Table of Values:

x
y
-4
0.063
-3
0.25
-2
1
-1
4

Now let's plot the points:

Graph:

Now draw a curve through all of the points to graph :


Graph of

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I'm going to use the quadratic formula:


Start with the given equation.


Add 0 to the left side (this does NOT change the 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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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 1. This means that to go from point to point, we can go down 2 and over 1



So starting at , go down 2 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


Now notice that the graph extends in both directions along the x-axis. So this means that ANY value of "x" can be plugged into the function.

So the domain is all real numbers.


Also, take note that the graph extends in both directions along the y-axis as well. So this tells us that the range is also ANY number.

So the range is all real numbers.

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We basically have this triangle set up:





Since the legs are and this means that and


Also, since the hypotenuse is , this means that .


Start with the Pythagorean theorem.


Plug in , ,


Square to get .


FOIL


Subtract 169 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 .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the possible answers are or


However, a negative length is NOT possible. So is NOT a solution


So the only answer is





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


Answer:


So the solution is


This means that the other leg is


So the two legs are: 12 and 5 units

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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).

You can put this solution on YOUR website!

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

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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


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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


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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

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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


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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