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Recall, the z score can be found by the formula




where is the mean and is the standard deviation.


So all you have to do is plug in the corresponding values to get:





and solve for to get the standard deviation.

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Start with the given equation.


Distribute.


Multiply.


Combine like terms on the right side.


Add to both sides.


Add to both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


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

Answer:

So the solution is which approximates to .

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Start with the given expression.


Distribute


Multiply the coefficients


Multiply the variable terms by adding the exponents.


Add


So

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Is the equation ???


Start with the given equation.


Multiply both sides by 5.


Multiply.


Add to both sides.


Combine like terms on the right side.


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

Answer:

So the solution is

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Remember, the perimeter of a square is simply the sum of the four sides of a square (ie add up ALL of the 4 sides). To shorten it, we can multiply one side by 4 (since they're all equal). So if "s" is the side length, then is the perimeter.



Start with the given equation.


Plug in the given perimeter


Divide both sides by 4 to isolate "s".


Reduce



So the side length is which means that the side of the square is 8 inches long.


Now that you have the side length, do you think you can continue? If not, repost or email me.

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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:.


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


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


Now combine like terms.


So FOILs to .


In other words, .

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From we can see that , , and


Start with the discriminant formula.


Plug in , , and


Square to get


Multiply to get


Subtract from to get


Since the discriminant is less than zero, this means that there are two complex solutions.


In other words, there are no real solutions.

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Start with the given expression.


Rewrite "1" as


Note: where .


Rewrite the second log using the identity


Combine the logs using the identity


So where

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Start with the given equation.


Distribute.


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 solution is

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Start with the second equation.


Multiply both sides by 2.


Multiply


Rearrange the equation


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


Move onto the first equation


Plug in


Add


Multiply both sides by 2.


Multiply


Divide both sides by to isolate .


Reduce.


So one of the numbers is 2


Go back to the second equation


Plug in


Multiply


So the other number is 12

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Start with the given equation.


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


Simplify


FOIL


Distribute


Get all terms to the left side.


Combine like terms.


Notice we have a quadratic 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 solutions are or

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Start with the given expression.


Multiply the first fraction by the reciprocal of the second fraction .


Factor to get .


Factor to get .


Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


Simplify.


Multiply


So simplifies to .


In other words, where or

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Start with the given expression.


Distribute


Group like terms


Combine like terms


So

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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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I'm assuming that you want to perform a partial fraction decomposition...


Start with the given expression.


Factor the denominator



Since both factors are linear and are of multiplicity 1, this means that the fraction decomposes into:




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


Simplify


Distribute


Rearrange the terms


Combine like terms.



Since the "y" coefficient on the right side is , this means that . Also, the constant term



So this means that we have the system:




Solve this system of equations (using any method) to get the solutions:

and


Now plug these values back into the original equation to get:






And simplify




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

Answer:

So

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# 1

Start with the given equation.


Get all terms to the left side.


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


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Simplify the square root


or Break up the expression.


So the answers are or

---

# 2

Start with the given inequality


Set the left side equal to zero


Set each individual factor equal to zero:

, or

Solve for x in each case:

, or


So our critical values are , and

Now set up a number line and plot the critical values on the number line





So let's pick some test points that are near the critical values and evaluate them.


Let's pick a test value that is less than (notice how it's to the left of the leftmost endpoint):

So let's pick

Start with the given inequality


Plug in


Evaluate and simplify the left side

Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.


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



Let's pick a test value that is in between and :

So let's pick

Start with the given inequality


Plug in


Evaluate and simplify the left side

Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.
So part our solution in interval notation is ()





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



Let's pick a test value that is in between and :

So let's pick

Start with the given inequality


Plug in


Evaluate and simplify the left side

Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.


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



Let's pick a test value that is greater than (notice how it's to the right of the rightmost endpoint):

So let's pick

Start with the given inequality


Plug in


Evaluate and simplify the left side

Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.
So part our solution in interval notation is ()





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





Summary:

So the solution in interval notation is:


() ()

---

# 3

Start with the given expression.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when

---

# 4




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:. Note: simply add the exponents


Multiply the Outer terms:.


Multiply the Inner terms:.


Multiply the Last terms:.


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


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


So FOILs to .


In other words, .

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The "x" is simply a place holder for any number. For example, when we say , we're saying: "take any number and add 5 to it".





Start with the given expression


Group like terms


Combine like terms


So

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Start with the given equation.


Multiply EVERY term by the LCD 12 to clear out the fractions.


Simplify


Multiply


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 solution is

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First, let's complete the square:


Start with the left side of the given equation.


Take half of the coefficient to get . In other words, .


Now square to get . In other words,


Now add and subtract . Make sure to place this after the "x" term. Notice how . So the expression is not changed.


Group the first three terms.


Factor to get .


Combine like terms.


So after completing the square, transforms to . So .


So is equivalent to .


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

Now let's solve


Start with the given equation.


Add to both sides.


Combine like terms.


Take the square root of both sides.


or Break up the "plus/minus" to form two equations.


or Simplify the square root.


or Subtract from both sides.


or Combine the fractions.


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


Answer:


So the solutions are or .

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3004 can be rounded to 3000 and 5982 rounded to 6000 (both are rounded to the nearest thousand)


So we can then multiply 3000 and 6000 mentally to get 18,000,000 (3*6 = 18 and just add 2 sets of 3 zeros)


So 3004 x 5982 estimates to 18,000,000

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Start with the given equation.


Subtract 5 from both sides.


Combine like terms.


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

Now let's complete the square for the left side expression


Start with the given expression.


Factor out the coefficient . This step is very important: the coefficient must be equal to 1.


Take half of the coefficient to get . In other words, .


Now square to get . In other words,


Now add and subtract inside the parenthesis. Make sure to place this after the "x" term. Notice how . So the expression is not changed.


Group the first three terms.


Factor to get .


Combine like terms.


Distribute.


Multiply.


So after completing the square, transforms to . So .


So is equivalent to .


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


Now let's solve


Start with the given equation.


Add to both sides.


Combine like terms.


Divide both sides by .


Reduce.


Take the square root of both sides.


or Break up the "plus/minus" to form two equations.


or Take the square root of to get .


or Add to both sides.


or Combine like terms.


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


Answer:


So the solutions are or .

  
__1__          ___1__         _1___ 
  3              9             27 

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The numerators follow the pattern: 1, 1, 1, ...
The denominators follow the pattern: 3, 9, 27, ...



Notice how if multiply 3 by 3, we get 9. If we multiply 9 by 3, we get 27, etc..


So the rule is to multiply the denominators by 3 to get the next term. So the next three denominators are: , , and


This means that the next three terms are: , and

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Start with the given equation.


Subtract 2 from both sides.


Combine like terms.


Now let's complete the square for the expression on the left side:



Start with the given expression.


Factor out the coefficient . This step is very important: the coefficient must be equal to 1.


Take half of the coefficient to get . In other words, .


Now square to get . In other words,


Now add and subtract inside the parenthesis. Make sure to place this after the "x" term. Notice how . So the expression is not changed.


Group the first three terms.


Factor to get .


Combine like terms.


Distribute.


Multiply.


So after completing the square, transforms to . So .


So is equivalent to .


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


Now let's solve


Start with the given equation.


Add to both sides.


Combine like terms.


Divide both sides by .


Reduce.


Take the square root of both sides.


or Break up the "plus/minus" to form two equations.


or Simplify the square root.


or Subtract from both sides.


or Combine the fractions.


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


Answer:


So the solutions are or .

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Note: the previous solution has a sign error and the answer should be much simpler.



Start with the given expression.


Multiply the first fraction by the reciprocal of the second fraction .


Factor to get .


Factor to get .


Factor to get .


Factor to get .


Combine the fractions.


Multiply and to get .


Highlight the common terms.


Cancel out the common terms.


Simplify


FOIL



So simplifies to .


In other words,

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Start with the given equation.


Multiply both sides by 10 to clear out the decimals.


Distribute and multiply.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the solution is

You can put this solution on YOUR website!

Start with the given equation.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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

Answer:

So the solution is

You can put this solution on YOUR website!

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 solution is

You can put this solution on YOUR website!
Is the expression ??? If it is, then...




Start with the given expression.


Multiply the first fraction by the reciprocal of the second fraction .


Factor to get .


Factor to get .


Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where , , or

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To solve this, let's first complete the square:


Start with the left side of the given equation.


Take half of the coefficient to get . In other words, .


Now square to get . In other words,


Now add and subtract . Make sure to place this after the "x" term. Notice how . So the expression is not changed.


Group the first three terms.


Factor to get .


Combine like terms.


So after completing the square, transforms to . So .


So is equivalent to .



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

Now let's solve


Start with the given equation.


Add to both sides.


Combine like terms.


Take the square root of both sides.


or Break up the "plus/minus" to form two equations.


or Take the square root of to get .


or Subtract from both sides.


or Combine like terms.


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


Answer:


So the solutions are or .

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# 1
If the equation is , then you are correct, the vertical asymptote is and the horizontal asymptote is (you're first choice)


# 2
Likewise, if the equation is , the vertical asymptote is and the horizontal asymptote is (again, you're first choice)


So you are correct on both of them.

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Standard Deviation:

First sort the numbers into ascending order (from least to greatest):
8,9,9,11,11,12,12,13,13



Now let's find the mean:



To find the mean, add up all of the numbers and divide the sum by the number of numbers (which in this case is 9).


So the mean is 10.8888888888889


Now let's find the standard deviation.


where is the arithmetic mean, is the ith number, and is the number of numbers

So we can replace N with 9 (since there are 9 numbers)



Replace with 10.8888888888889



Expand the summation (replace each with the respective number)




Subtract the terms in the parenthesis



Square each term



Add up all of the terms



Multiply



Take the square root



So the standard deviation is


Since and is the variance, this means that the variance is: