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Start with the given expression.


Plug in and


Square 2 to get 4.


Multiply


Reduce.


So when and

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Start with the given expression.


Expand


FOIL.


Multiply.


Replace with . Note: .


Multiply.


Combine like terms.


So .

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Start with the given equation.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


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Answer:

So the solution is

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Hint: Solve the proportion .

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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 = 30
2*15 = 30
3*10 = 30
5*6 = 30
(-1)*(-30) = 30
(-2)*(-15) = 30
(-3)*(-10) = 30
(-5)*(-6) = 30

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=31
2	15	2+15=17
3	10	3+10=13
5	6	5+6=11
-1	-30	-1+(-30)=-31
-2	-15	-2+(-15)=-17
-3	-10	-3+(-10)=-13
-5	-6	-5+(-6)=-11

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 .


In other words, .


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

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Since "there were 2 times as many children's tickets sold as adult tickets", this means that . Let's say that there were x=10 adult tickets. This then means that there are y=2(10)=20 children's tickets since "there were 2 times as many children's tickets sold as adult tickets".


So the third equation is . If you get everything to one side, you get

So the three equations are




Note: I multiplied ALL of the terms of the second equation by 10 to make every number a whole number.


which translates into the matrix equation




Now append the right side to the 3x3 matrix on the left side to get the augmented matrix




Now let's row reduce. Note: Solution provided by the Linear Algebra Toolkit




Now look at the last column of the last matrix (ie the last step shown). This column has the values 175, 350, and 110. This means that , and


So 175 adults, 350 children, and 110 seniors went to the movies that day.

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Hint: Find the slope of the line segments AB, BC, CD, and AD. If you can show that the slope of AB is equal to the slope of CD and show that the slope of BC is equal to the slope of AD, then you will have shown that ABCD is a parallelogram.

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Let . Now convert to exponential form to get . We can rewrite 64 as and 4 as which means that and . Since the bases are equal, the exponents must be equal. So . Solve for 'y' to get . This then means that

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So

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Start with the given inequality.


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 . note: Remember, the inequality sign flips when we divide both sides by a negative number.


Reduce.


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

Answer:

So the solution is

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One easy way to find a number between two numbers is to simply average the two given numbers. Ie add them up and divide by 2:




So the number , which is the mixed number , is in between the numbers and


Other ways would involve you picking a number between 9 and 10, say 9.7, and writing that as a fraction over 8 like this:

So , which is the mixed number , is another number in between the numbers and

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1. which means that

2. which means that

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Start with the given equation.


Multiply EVERY term by the LCD to clear the fractions.


Multiply and simplify.


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.


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

Answer:

So the solution is

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Start with the given inequality.


Distribute.


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


The answer in set-builder notation is

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Only post one problem at a time please.

# 1

which means that

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Start with the given equation.


Subtract from both sides.


Combine like terms on the right side.


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

Answer:

So the solution is

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Start with the given inequality.


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 given equation.


Cross multiply.


Multiply 2 and 4 to get 8.


Distribute.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


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

Answer:

So the solution is

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. So

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Solved by pluggable solver: FOIL (ie Expand) Two Binomials

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 --------------------- Answer: So FOILs and simplifies to In other words,

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which means that

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Hint: If 'a' and 'b' are solutions of a quadratic equation, then

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Start with the given equation.


Cross multiply.


Multiply.


Take the square root of both sides.


or Break up the plus/minus.


or Take the square root of 36 to get 6.


So the solutions are or

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Start with the given equation.


Notice that the quadratic is 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)


Break up the fraction.


Reduce.


or Break up the expression.


So the solutions are or

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Hint: Since "The product of page numbers on two facing pages of a book is 182", this means that where 'x' is the first of the two pages and 'x+1' is the next page.

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Hint: Simply replace 'x' with -1 and compute the right side.

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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


In other words,


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# 2



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 = 14
2*7 = 14
(-1)*(-14) = 14
(-2)*(-7) = 14

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=15
2	7	2+7=9
-1	-14	-1+(-14)=-15
-2	-7	-2+(-7)=-9

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 .


In other words, .


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

------------------------------------------------------------------------
# 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 :
1,2,3,4,6,8,12,16,24,48
-1,-2,-3,-4,-6,-8,-12,-16,-24,-48


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


These factors pair up and multiply to .
1*(-48) = -48
2*(-24) = -48
3*(-16) = -48
4*(-12) = -48
6*(-8) = -48
(-1)*(48) = -48
(-2)*(24) = -48
(-3)*(16) = -48
(-4)*(12) = -48
(-6)*(8) = -48

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	-48	1+(-48)=-47
2	-24	2+(-24)=-22
3	-16	3+(-16)=-13
4	-12	4+(-12)=-8
6	-8	6+(-8)=-2
-1	48	-1+48=47
-2	24	-2+24=22
-3	16	-3+16=13
-4	12	-4+12=8
-6	8	-6+8=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 .


In other words, .


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

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# 1
Start with the given equation

Now set each factor equal to zero:

or

Now solve for b for each factor:

or

So the solutions are or

----------------------------------------------------------------
# 2
Start with the given equation


Factor the left side


Now set each factor equal to zero:
or

or Now solve for b in each case


So the solutions are or

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It's not much of a simplification (in my opinion) as the expression doesn't get any simpler, but here it goes...


Start with the given expression.


Divide both the numerator and the denominator by 'c'


Reduce to get 1.


Rewrite as . Note: this implies that 'c' is a non-negative number, which it is.


Combine the lower square roots using the identity


Break up the lower inner fraction.


Reduce to get 1.


So

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So where

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


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


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

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	20	1+20=21
2	10	2+10=12
4	5	4+5=9
-1	-20	-1+(-20)=-21
-2	-10	-2+(-10)=-12
-4	-5	-4+(-5)=-9

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 .


In other words, .


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