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note: where . In other words, any number to the zeroth power (except zero) is 1.

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Start with the given expression


Factor out the GCF


Now let's focus on the inner expression




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





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 :
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36


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


These factors pair up and multiply to .
1*36
2*18
3*12
4*9
6*6
(-1)*(-36)
(-2)*(-18)
(-3)*(-12)
(-4)*(-9)
(-6)*(-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	36	1+36=37
2	18	2+18=20
3	12	3+12=15
4	9	4+9=13
6	6	6+6=12
-1	-36	-1+(-36)=-37
-2	-18	-2+(-18)=-20
-3	-12	-3+(-12)=-15
-4	-9	-4+(-9)=-13
-6	-6	-6+(-6)=-12

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


Condense


So factors to


This means that the expression goes from and factors further to



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

So factors to

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# 1



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

---

# 2


Looking at we can see that the first term is and the last term is where the coefficients are 1 and -2 respectively.

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



Factors of -2:
1,2

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

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

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	-2	1+(-2)=-1
-1	2	-1+2=1

From this list we can see that -1 and 2 add up to 1 and multiply to -2


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

---

# 3



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


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


These factors pair up and multiply to .
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)

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	-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 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 equation.


Distribute the negative


Group the common terms


Combine like terms



So

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Do you want to factor this?



Start with the given expression.


Rewrite as .


Rewrite as .


Notice how we have a difference of squares. So let's use the difference of squares formula to factor the expression:


Factor the expression using the difference of squares.


So factors to .


In other words .

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Start with the given equation.


Multiply both sides by the LCD to clear any fractions.


Distribute and multiply.


Distribute again.


Combine like terms on the left side.


Add to 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 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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So the equation is ?


Start with the given equation.


Divide both sides by 180.


Subtract 1 from both sides.


Combine the fractions.


Multiply both sides by "n".


Multiply both sides by 180.


Multiply


Divide both sides by .

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Note: $0.25 = 25 cents and $1 = 100 cents


Start with the given formula.


Plug in 25 cents for the numerator and 100 cents for the denominator


Cancel like terms


Simplify


So $0.25 is 25% of one U.S. dollar

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Let x=# of $6 movies and y=# of $4 movies

Since "you have watched a total of 10 movies", this means that the first equation is

Also, since you only have $50 to spend, this means that the second equation is



So we have the system of equations:




Let's use substitution to solve this system.

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




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

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 40 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



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


So the second part of our answer is:









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

So our answers are:

and


This means that you bought 5 $6 movies and 5 $4 movies

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What do you want to do with this equation? Do you want to solve for n?

Start with the given equation.


Subtract 20 from both sides.


Divide both sides by 2.


Rearrange the equation

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Start with the given expression.


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:.


Now collect every term to make a single expression.


Now combine like terms.


So FOILs to .


In other words, .

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Start with the given expression.


Factor to get . Notice how the LCD is


Multiply the first term by . The goal is to get the first denominator equal to the LCD


Combine the fractions.


Distribute


Add the fractions.


Combine like terms.


So where or

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Start with the given inequality


Plot the point on a number line




Now plug in into the inequality





Since is true, this means that we shade the entire region that is in.


So shade to the right of . Note: there is a open circle at

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# 1



Start with the given expression.


Divide the coefficients and to get


When you divide monomials, simply subtract the exponents.


Subtract.


Rewrite as


Combine the fractions.


So where or

---

# 2

Note: and


So where and

---

# 3



So

---

# 4




So

---

# 5



So

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Start with the given expression.


Factor to get .


Factor to get .


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where or

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Perimeter of a triangle:

P=side1+side2+side3

P=6+8+10=14+10=24


So the perimeter of the triangle is 24 inches

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Notice how as x goes up by 1, y goes up by 3. So this implies a linear relationship. This means that the equation is a line.



So let's find equation of the line through the first two points (0,-2) and (1,1) (note: you can pick any pair of points)




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


Start with the slope formula.


Plug in , , , , ,


Subtract from to get


Subtract from to get


Reduce


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.


Simplify


So the equation that goes through the points and is


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

Answer:

So the function takes the x values and produces the y values is


Notice how the graph of goes through the points (0,-2), (1,1), (2,4), (3,7) and (4,10). So this visually verifies our answer.


Graph of

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Start with the slope formula.


Plug in , , , , ,


Subtract from to get


Subtract from to get


Reduce


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

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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.


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


Distribute


Multiply


Add 3 to both sides.


Combine like terms.


So the equation of the line parallel to that goes through the point is .


Here's a graph to visually verify our answer:
Graph of the original equation (red) and the parallel line (green) through the point .

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Two lines are parallel if and only if their slopes are equal.

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Surface Area:

Solved by pluggable solver: To calculate Surface area of a sphere

Geometry Surface Area and Volume of 3 Dimensional Figures The Formula for Surface area Sphere is Hence, The Surface Area of Sphere of radius 14 is 2463.0086376. For more on this topic, See the lessons on Geometry: Area and Surface Area Some more is on Surface Area and Volume and Perimeter and Area.

Volume:

Solved by pluggable solver: FIND volume of a sphere

Volume of a sphere

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Solved by pluggable solver: Surface area and Volume of a cylinder

The surface area (SA) of a cylinder, excluding the base area and top area, is given by When the areas of base and top are included, the surface area becomes The volume of the cone can be found by the formula In our case the desired values are thus: SA (without base and top) = SA (with base and top) = V = For more on this topic, refer to Surface area and Volume.

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[-(-2x^2)^2]^3 = [-(-2x^2)(-2x^2)]^3 
               = [-(4x^4)]^3
               = [-4x^4]^3
               = -64x^12

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-4[5x-6(-5+x)] = -4[5x+30-6x]
               = -4[-x+30]
               = 4x-120

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Let's simplify this expression using synthetic division


Start with the given expression

First lets find our test zero:

Set the denominator equal to zero

Solve for x.

so our test zero is -2


Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.

-2	|	2	2	-3	2
	|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 2)

-2	|	2	2	-3	2
	|
		2

Multiply -2 by 2 and place the product (which is -4) right underneath the second coefficient (which is 2)

-2	|	2	2	-3	2
	|		-4
		2

Add -4 and 2 to get -2. Place the sum right underneath -4.

-2	|	2	2	-3	2
	|		-4
		2	-2

Multiply -2 by -2 and place the product (which is 4) right underneath the third coefficient (which is -3)

-2	|	2	2	-3	2
	|		-4	4
		2	-2

Add 4 and -3 to get 1. Place the sum right underneath 4.

-2	|	2	2	-3	2
	|		-4	4
		2	-2	1

Multiply -2 by 1 and place the product (which is -2) right underneath the fourth coefficient (which is 2)

-2	|	2	2	-3	2
	|		-4	4	-2
		2	-2	1

Add -2 and 2 to get 0. Place the sum right underneath -2.

-2	|	2	2	-3	2
	|		-4	4	-2
		2	-2	1	0

Since the last column adds to zero, we have a remainder of zero. This means is a factor of

Now lets look at the bottom row of coefficients:

The first 3 coefficients (2,-2,1) form the quotient




So

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Start with the given equation.


Set the right side greater than zero.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is which means that any value of x that is greater than 4 will make greater than zero.

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Start with the given equation.


Set the right side greater than zero.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the answer is which means that any value of x that is greater than 4 will make greater than zero.

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Start with the first equation.


Plug in


Multiply both sides by the LCD to clear any fractions.


Distribute and multiply.


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. This is the first part of our answer.


Move onto the second equation


Plug in x=3


Simplify. This is the second part of our answer.


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

Answer:

So the solutions are and which form the ordered pair (3,-2)

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Remember,

Profit = Revenue - Cost

which is symbolically:





Plug in and


Distribute the negative.


Combine like terms


Now the break even point occurs when you neither gain money nor lose money. In other words, break even point happens when the profit is zero. So this means that the break even point occurs when


Plug in


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


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


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


or Break up the expression.


So the answers are or


which approximate to or


So the break even points occur when the price is either $4.33 or $11.67