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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.


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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

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# 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).

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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 :
1,2,4,5,8,10,20,25,40,50,100,200
-1,-2,-4,-5,-8,-10,-20,-25,-40,-50,-100,-200


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


These factors pair up and multiply to .
1*(-200) = -200
2*(-100) = -200
4*(-50) = -200
5*(-40) = -200
8*(-25) = -200
10*(-20) = -200
(-1)*(200) = -200
(-2)*(100) = -200
(-4)*(50) = -200
(-5)*(40) = -200
(-8)*(25) = -200
(-10)*(20) = -200

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	-200	1+(-200)=-199
2	-100	2+(-100)=-98
4	-50	4+(-50)=-46
5	-40	5+(-40)=-35
8	-25	8+(-25)=-17
10	-20	10+(-20)=-10
-1	200	-1+200=199
-2	100	-2+100=98
-4	50	-4+50=46
-5	40	-5+40=35
-8	25	-8+25=17
-10	20	-10+20=10

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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Start with the given expression


Factor to get


Factor out the GCF


So factors to


In other words,

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Start with the given equation.


Add 4 to both sides.


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


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 solutions are or

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which means that

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We basically have this triangle set up:





To find the unknown length, we need to use the Pythagorean Theorem.


Remember, the Pythagorean Theorem is where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.


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 .


Square to get .


Combine like terms.


Rearrange the equation.


Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).


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


Answer:


So the solution is which approximates to .

So the ramp is about 43.186 ft long.

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


Note: is the first point . So this means that and .
Also, is the second point . So this means that and .


Start with the slope formula.


Plug in , , , and


Subtract from to get


Subtract from to get


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


Rewrite as


Distribute


Multiply


Subtract 2 from both sides.


Combine like terms. note: If you need help with fractions, check out this solver.


So the equation that goes through the points and is

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I'm assuming that the equation is really


Start with the given equation.


Plug in


Multiply -4 and 1 to get -4


Add -4 to 5 to get 1.


So when , the value of y is

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Using the identity , where 'x' is a real number, this means that

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Hint: and

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Start with the given expression


Rearrange the terms.


Group the 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


Rearrange the terms.


So factors to


In other words,

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Note: is the first point . So this means that and .
Also, is the second point . So this means that and .


Start with the distance formula.


Plug in , , , and .


Subtract from to get .


Subtract from to get .


Square to get .


Square to get .


Add to to get .


Simplify the square root.


So our answer is


Which approximates to


So the distance between the two points is approximately 6.708 units.

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Looking at we can see that the first term is and the last term is where the coefficients are 3 and -16 respectively.

Now multiply the first coefficient 3 and the last coefficient -16 to get -48. Now what two numbers multiply to -48 and add to the middle coefficient 2? Let's list all of the factors of -48:



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

-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...List the negative factors as well. This will allow us to find all possible combinations

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

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2

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 this list we can see that -6 and 8 add up to 2 and multiply to -48


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


In other words,

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Start with the given expression


Group the 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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Start with the given expression


Rearrange the terms.


Group the 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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Start with the given expression.


Rewrite as . Rewrite as .


Now factor by using the sum of cubes formula. Remember the sum of cubes formula is


Multiply

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

So factors to .

In other words,

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Looking at we can see that the first term is and the last term is where the coefficients are 3 and 8 respectively.

Now multiply the first coefficient 3 and the last coefficient 8 to get 24. Now what two numbers multiply to 24 and add to the middle coefficient -10? Let's list all of the factors of 24:



Factors of 24:
1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

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

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


Now which of these pairs add to -10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -10

First Number	Second Number	Sum
1	24	1+24=25
2	12	2+12=14
3	8	3+8=11
4	6	4+6=10
-1	-24	-1+(-24)=-25
-2	-12	-2+(-12)=-14
-3	-8	-3+(-8)=-11
-4	-6	-4+(-6)=-10

From this list we can see that -4 and -6 add up to -10 and multiply to 24


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