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If one of the angles is x degrees, then the other acute angle is degrees.

You can put this solution on YOUR website!
The expression is undefined when the denominator . Solve for x to get ---->



So if , then is true. So if , the expression is undefined.


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You can put this solution on YOUR website!















So when


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So the solution is


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Well, you can multiply every term by the LCD to get





and then you can isolate z by dividing both sides by to get





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I'll do the first one to get you started.

# 1

Start with the given equation.


Distribute.


Combine like terms on the left side.


Subtract from 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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I'm assuming you want to factor this.




Start with the given expression.


Factor out the GCF .


Now let's try to factor 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 :
1,3,9
-1,-3,-9


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


These factors pair up and multiply to .
1*9 = 9
3*3 = 9
(-1)*(-9) = 9
(-3)*(-3) = 9

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	9	1+9=10
3	3	3+3=6
-1	-9	-1+(-9)=-10
-3	-3	-3+(-3)=-6

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


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


So then factors further to


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


Answer:


So completely 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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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 coefficient is .


Now multiply the first coefficient by the last coefficient 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 :
1,3,11,33
-1,-3,-11,-33


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


These factors pair up and multiply to .
1*(-33) = -33
3*(-11) = -33
(-1)*(33) = -33
(-3)*(11) = -33

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	-33	1+(-33)=-32
3	-11	3+(-11)=-8
-1	33	-1+33=32
-3	11	-3+11=8

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 the GCF from the second group.


Factor out from the entire expression.


So completely factors to


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You can put this solution on YOUR website!
I'm assuming you want to factor this.




Start with the given expression.


Factor out the GCF .


Now let's try to factor 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 :
1,2,3,4,5,6,8,9,10,12,15,16,18,19,20,24,30,32,36,38,40,45,48,57,60,72,76,80,90,95,96,114,120,144,152,160,171,180,190,228,240,285,288,304,342,360,380,456,480,570,608,684,720,760,855,912,1140,1368,1440,1520,1710,1824,2280,2736,3040,3420,4560,5472,6840,9120,13680,27360
-1,-2,-3,-4,-5,-6,-8,-9,-10,-12,-15,-16,-18,-19,-20,-24,-30,-32,-36,-38,-40,-45,-48,-57,-60,-72,-76,-80,-90,-95,-96,-114,-120,-144,-152,-160,-171,-180,-190,-228,-240,-285,-288,-304,-342,-360,-380,-456,-480,-570,-608,-684,-720,-760,-855,-912,-1140,-1368,-1440,-1520,-1710,-1824,-2280,-2736,-3040,-3420,-4560,-5472,-6840,-9120,-13680,-27360


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


These factors pair up and multiply to .
1*(-27360) = -27360
2*(-13680) = -27360
3*(-9120) = -27360
4*(-6840) = -27360
5*(-5472) = -27360
6*(-4560) = -27360
8*(-3420) = -27360
9*(-3040) = -27360
10*(-2736) = -27360
12*(-2280) = -27360
15*(-1824) = -27360
16*(-1710) = -27360
18*(-1520) = -27360
19*(-1440) = -27360
20*(-1368) = -27360
24*(-1140) = -27360
30*(-912) = -27360
32*(-855) = -27360
36*(-760) = -27360
38*(-720) = -27360
40*(-684) = -27360
45*(-608) = -27360
48*(-570) = -27360
57*(-480) = -27360
60*(-456) = -27360
72*(-380) = -27360
76*(-360) = -27360
80*(-342) = -27360
90*(-304) = -27360
95*(-288) = -27360
96*(-285) = -27360
114*(-240) = -27360
120*(-228) = -27360
144*(-190) = -27360
152*(-180) = -27360
160*(-171) = -27360
(-1)*(27360) = -27360
(-2)*(13680) = -27360
(-3)*(9120) = -27360
(-4)*(6840) = -27360
(-5)*(5472) = -27360
(-6)*(4560) = -27360
(-8)*(3420) = -27360
(-9)*(3040) = -27360
(-10)*(2736) = -27360
(-12)*(2280) = -27360
(-15)*(1824) = -27360
(-16)*(1710) = -27360
(-18)*(1520) = -27360
(-19)*(1440) = -27360
(-20)*(1368) = -27360
(-24)*(1140) = -27360
(-30)*(912) = -27360
(-32)*(855) = -27360
(-36)*(760) = -27360
(-38)*(720) = -27360
(-40)*(684) = -27360
(-45)*(608) = -27360
(-48)*(570) = -27360
(-57)*(480) = -27360
(-60)*(456) = -27360
(-72)*(380) = -27360
(-76)*(360) = -27360
(-80)*(342) = -27360
(-90)*(304) = -27360
(-95)*(288) = -27360
(-96)*(285) = -27360
(-114)*(240) = -27360
(-120)*(228) = -27360
(-144)*(190) = -27360
(-152)*(180) = -27360
(-160)*(171) = -27360

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	-27360	1+(-27360)=-27359
2	-13680	2+(-13680)=-13678
3	-9120	3+(-9120)=-9117
4	-6840	4+(-6840)=-6836
5	-5472	5+(-5472)=-5467
6	-4560	6+(-4560)=-4554
8	-3420	8+(-3420)=-3412
9	-3040	9+(-3040)=-3031
10	-2736	10+(-2736)=-2726
12	-2280	12+(-2280)=-2268
15	-1824	15+(-1824)=-1809
16	-1710	16+(-1710)=-1694
18	-1520	18+(-1520)=-1502
19	-1440	19+(-1440)=-1421
20	-1368	20+(-1368)=-1348
24	-1140	24+(-1140)=-1116
30	-912	30+(-912)=-882
32	-855	32+(-855)=-823
36	-760	36+(-760)=-724
38	-720	38+(-720)=-682
40	-684	40+(-684)=-644
45	-608	45+(-608)=-563
48	-570	48+(-570)=-522
57	-480	57+(-480)=-423
60	-456	60+(-456)=-396
72	-380	72+(-380)=-308
76	-360	76+(-360)=-284
80	-342	80+(-342)=-262
90	-304	90+(-304)=-214
95	-288	95+(-288)=-193
96	-285	96+(-285)=-189
114	-240	114+(-240)=-126
120	-228	120+(-228)=-108
144	-190	144+(-190)=-46
152	-180	152+(-180)=-28
160	-171	160+(-171)=-11
-1	27360	-1+27360=27359
-2	13680	-2+13680=13678
-3	9120	-3+9120=9117
-4	6840	-4+6840=6836
-5	5472	-5+5472=5467
-6	4560	-6+4560=4554
-8	3420	-8+3420=3412
-9	3040	-9+3040=3031
-10	2736	-10+2736=2726
-12	2280	-12+2280=2268
-15	1824	-15+1824=1809
-16	1710	-16+1710=1694
-18	1520	-18+1520=1502
-19	1440	-19+1440=1421
-20	1368	-20+1368=1348
-24	1140	-24+1140=1116
-30	912	-30+912=882
-32	855	-32+855=823
-36	760	-36+760=724
-38	720	-38+720=682
-40	684	-40+684=644
-45	608	-45+608=563
-48	570	-48+570=522
-57	480	-57+480=423
-60	456	-60+456=396
-72	380	-72+380=308
-76	360	-76+360=284
-80	342	-80+342=262
-90	304	-90+304=214
-95	288	-95+288=193
-96	285	-96+285=189
-114	240	-114+240=126
-120	228	-120+228=108
-144	190	-144+190=46
-152	180	-152+180=28
-160	171	-160+171=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


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


So then factors further to


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


Answer:


So completely 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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There are 9! = 9*8*7*6*5*4*3*2*1 = 362880 different ways to distribute the 9 pens to the 9 students.


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Start with the given equation.


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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You can put this solution on YOUR website!
You made a mistake in thinking that multiplies out to ; however, it really multiplies out to


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 "y":


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 solution is


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You can put this solution on YOUR website!



So


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You can put this solution on YOUR website!

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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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 .


Take the square root of to get .


So our answer is


So the distance between the two points is 10 units.


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P(Both Vowels) = P(Vowel from PRIME) * P(Vowel from MATH)

P(Both Vowels) = (2/5)*(1/4)

P(Both Vowels) = 2/20

P(Both Vowels) = 1/10


So the probability is

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Multiply both sides by 13 to get which becomes


So the solution is


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There are ways to toss 4 coins, but only 1 way to toss all 4 heads. So the probability is

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In general, the slope of is m and the y-intercept is (0,b). Since m=3, the slope is 3. Because b=-4, the y-intercept is (0,-4)


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The weights determine how important a given subject is. So if a class has a higher weight, it will factor more towards the average grade. This is why you can't just average the given marks like normal.


Instead, you simply multiply each weight by its corresponding grade and then you divide by 11 (the sum of the weights) to get


(3*89+3*92+2*94+2*81+1*85)/11 = (267+276+188+162+85)/11 = 978/11 = 88.909



So the average is 88.909


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Simply add 3 to both sides to get which simplifies to


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"the product of five more than a number and six is thirty" translates to


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Let L = Leah's age, S = Sue's age, and J = John's age


So because "Leah is 6 years older than Sue", we know that L = S+6 (ie add 6 years to Sue's age to get Leah's age)


Since "John is 5 years older than Leah", we also know that J = L+5


Finally, we know that "the total of their ages is 41", which tells us that L + S + J = 41


So simply plug in J = L+5 to get

L + S + (L+5) = 41

L + S + L + 5 = 41


Now plug in L = S + 6 to get

(S+6) + S + (S+6) + 5 = 41

S + 6 + S + S + 6 + 5 = 41


Now let's solve for S


S + 6 + S + S + 6 + 5 = 41

3S + 17 = 41

3S = 41 - 17

3S = 24

S = 24/3

S = 8


So Sue is 8 years old (since S = 8 )


So Leah is L = S+6 = 8+6 = 14 years old (since L = 14)


And John is J = L + 5 = 14 + 5 = 19 years old.


Notice that the sum of their ages is


8+14+19 = 22+19 = 41


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You can put this solution on YOUR website!
Let L = Leah's age, S = Sue's age, and J = John's age


So because "Leah is 6 years older than Sue", we know that L = S+6 (ie add 6 years to Sue's age to get Leah's age)


Since "John is 5 years older than Leah", we also know that J = L+5


Finally, we know that "the total of their ages is 41", which tells us that L + S + J = 41


So simply plug in J = L+5 to get

L + S + (L+5) = 41

L + S + L + 5 = 41


Now plug in L = S + 6 to get

(S+6) + S + (S+6) + 5 = 41

S + 6 + S + S + 6 + 5 = 41


Now let's solve for S


S + 6 + S + S + 6 + 5 = 41

3S + 17 = 41

3S = 41 - 17

3S = 24

S = 24/3

S = 8


So Sue is 8 years old (since S = 8 )


So Leah is L = S+6 = 8+6 = 14 years old (since L = 14)


And John is J = L + 5 = 14 + 5 = 19 years old.


Notice that the sum of their ages is


8+14+19 = 22+19 = 41


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Hint: Given events A and B, P(A or B) = P(A) + P(B) - P(A and B)

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Perimeter = sum of 3 sides = 5 + 11 + 11 = 16 + 11 = 27


So the perimeter is 27 ft.


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You can put this solution on YOUR website!

Start with the given expression.


Factor out the GCF .


Now let's try to factor 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 :
1,2,3,6,9,18
-1,-2,-3,-6,-9,-18


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


These factors pair up and multiply to .
1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18

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	-18	1+(-18)=-17
2	-9	2+(-9)=-7
3	-6	3+(-6)=-3
-1	18	-1+18=17
-2	9	-2+9=7
-3	6	-3+6=3

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


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


So then factors further to


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


Answer:


So completely 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).


If you need more help, email me at jim_thompson5910@hotmail.com

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Jim

You can put this solution on YOUR website!
Since the diameter is 4 ft, the radius is 4/2=2 ft. So r=2


Start with the volume of a cylinder formula.


Plug in and .


Replace with (note: Use more digits of to get better accuracy).


Square to get .


Multiply , and to get (this value is approximate).


So the volume is approximately 502.654824 cubic ft.


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You can put this solution on YOUR website!



So


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Jim

You can put this solution on YOUR website!
Start with the given equation.


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


Subtract 3x from 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 .


or Break up the expression.


So the solutions are or


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You can put this solution on YOUR website!
The key here is to eliminate one variable so we can solve for the other. So I'm going to eliminate y.


ax-by=b+a
bx+ay=b+a



a^2x-aby=ab+a^2
b^2x+aby=b^2+ab

Note: I multiplied both sides of the first equation by 'a' and both sides of the second equation by 'b'


Now add the two equations to get


a^2x+b^2x=a^2+2ab+b^2


(a^2+b^2)x=a^2+2ab+b^2


x=(a^2+2ab+b^2)/(a^2+b^2)


Now that we know that x=(a^2+2ab+b^2)/(a^2+b^2), we can use it to solve for 'y'. I'll let you do that.