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You divide 972 by 4096 and then take the fifth root to find the common ratio r...since you need four intermediate geometric means, there will be five increments...when you do that you find that r = 3/4...so that your sequence looks like...
4096, 3072, 2304, 1728, 1296, 972

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The terms of the series are
1 + 2 + 4 + 8 + 16 + 32 =
63

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It is not a Pythagorean triple, since
8^2 + 10^2 ≠ 12^2

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Slope m is found by the formula
m = (y2 - y1) / (x2 - x1)
So for (-5,4) and (1,-2), we have
m = (-2 - 4) / (1 -(-5)) =
m = -6 / 6 = -1
And for (3,2) and (-5,0), we have
m = (0 - 2) / (-5 - 3) =
m = -2 / -8
m = 1/4

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This is a linear function showing a relationship between x and y.
For y = mx + c, we have y expressed as a function of x, whereby any increase in x is calculated to be an (m times x) increase in y. The c value is the value of y when x = 0.
If this function is graphed as a line, m is the slope of that line and c is the y-intercept.

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Well 2:30 pm represents three sigmas below the mean...so look up on your normal distribution table a z-score of -3.00 and find the probability of .0013.
Thus the probability is .13% that a student chosen at random arrives before 2:30 pm.

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We need a line perpendicular to
2x + 4y = -1
Let us find the slope of this line first...
4y = -2x - 1
y = (-1/2)x - 1/4
The slope here is -1/2.
The slope of the perpendicular to it is the negative reciprocal, or +2.
Now use the point-slope form
y - y1 = m(x - x1)
y - 3 = 2(x + 1)
y - 3 = 2x + 2
y = 2x + 5

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Since x^2 + 5x + 6 factors into (x + 2)(x + 3), thus
any pair of factors of (x + 2)(x + 3)(x + 5) could represent
the area of the base...just multiply them to get the polynomial...

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Your best bet with fractional equations to to multiply eveyrthing by the lowest common denominator in order to clear the fractions...so for
1/2x^2 + 5x + 25/2 = 0
we multiply by 2 and get
x^2 + 10x + 25 = 0
Now factor and solve as usual...
(x + 5)^2 = 0
x = -5 (double root)

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Well 21 discs one way and 11 disks the other way can be cut...231 disks in total...let's find thier total area and subtract it from the area of the original sheet...
A = 231 * ((pi)r^2)
A = 231 * 16pi
A = 3696*pi sq cm
The original sheet was 90 x 170 or 15300 sq cm.
Now subtract to get the leftover area
Leftover = 15300 - 3696(pi) or about 3689 sq cm

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I believe this one is done like this...
P(3 of 5 have reduced price) = C(5,3)*(1/4)^3*(3/4)^2 =
10 * 1/64 * 9/16 =
45/512

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If you mean
y = 2x + 3
then the slope is 2.

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In problems like this, we always use distance = rate * time, or D = RT.
Thus T = D/R. Let R be the plane's speed in still air.
Going against the wind, we have
T1 = 720 / (R - 30) and
T2 = 720 / (R + 30)
But these two times add to 10 hours, so we have
720 / (R - 30) + 720 / (R + 30) = 10
Now multiply everything by (R+30)(R-30) to clear fractions...
720(R+30) + 720(R-30) = 10(R+30)(R-30)
Now solve for R...
720R + 21600 + 720R - 21600 = 10(R^2 - 900)
1440R = 10(R^2 - 900)
144R = R^2 - 900
R^2 - 144R - 900 = 0
(R - 150)(R + 6) = 0
R = 150 mph

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Since the question is an OR case, we add the probabilities...
The probability of picking a king is 4/52 or 1/13 and
the probability of picking a red card is 26/52 or 1/2, we add
1/13 + 1/2 =
2/26 + 13/26 =
15/26
Thus 15/26 is the probability of any one card be a king or red.
To do it twice (after first replacing the card), one would then multiply
15/26 by itself, giving us 225/676.
To do it in either card, we add 15/26 + 15/26 = 15/13...or a certainty...
It is not clear how we draw the two cards.

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For x^2 - 3x - 12 = 0
we examine the discriminant as b^2 - 4ac...here that is
(-3)^2 - 4(1)(-12) =
9 + 48 =
57

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Yes it is.
6x/12x = 1/2
Either the book is wrong or you copied something incorrectly...

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Okay let their ages be x and x+1, since they are a year apart. Then we have
x^2 + (x+1)^2 + x + x+1 = 1105
Now solve this...
x^2 + x^2 + 2x + 1 + x + x + 1 = 1105
2x^2 + 4x + 2 = 1105
2x^2 + 4x - 1103 = 0
This is not factorable, so we use the quadratic formula to get
x = (-4 ± sqrt(16 + 8824)) / 4
x = (-4 ± sqrt(8840)) / 4
x = (-4 ± 2*sqrt(2210)) / 4
x = (-2 ± sqrt(2210)) / 2
x is about 22.5 years old
x+1 is about 23.5 years old
I have a feeling there is something wrong with the problem...

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Since we have a product equalling zero, the only way that can happen is if one of the numbers involved IS zero...thus for
8x(4.7) = 0
x must be zero.

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Okay, from
8x^2 = 5x
let's subtract the 5x and then factor giving us
8x^2 - 5x = 0
x(8x - 5) = 0
so that
x = 0 or 8x - 5 = 0
The second one can be solved giving us
8x = 5
x = 5/8 as the second solution...

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2x^3 - 10x^2 = 0
factor out 2x^2 and get
2x^2(x - 5) = 0
Therefore x = 0 and x = 5

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8 gal. 7 qt.
- 3 gal. 2 qt.
__________________
5 gal. 5 qt.
which equals
6 gal. 1 qt.

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Okay, we can express -125 as (-5)^3, so we have
(-125)^(-3/2) =
((-5)^3)^(-3/2) =
(-5)^(-9/2) =
1 / (-5)^(9/2) =
1 / [(-5)^4 * (-5)^(1/2)] =
1 / 625i*sqrt(5)
Now we have to rationalize by multiplying top and bottom by -i*sqrt(5) and we get
-i*sqrt(5) / 3125

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Okay...
(7/2)^2 means
(7/2) * (7/2) =
49/4 or 12 1/4

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From
y - 7 = 3x
we add 7 to both sides and get
y - 7 + 7 = 3x + 7
y = 3x + 7

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Let the numbers be x and y...then we have
x + y = 54 and
x - y = 18
Now add the equations and get
2x = 72 and
x = 36
From this you can see that
y = 18 and we're done...

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For every nut type, we have 6 fruit and 5 granola...thus 30 for each nut type and there are 8 nut types, so the answer turns out to be
8 * 6 * 5 =
240 mixes

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We can add up the possibilities...
start with having all three women...how many ways can we choose that?
Well 4 chosen three at a time is C(4,3) = 4 ways
Now, let's look at just two women...how many ways can we choose them? Well, C(4,2) = 6 ways...but for each of these ways, we can have any of six men, so there's a total of 36 ways this can happen...
Thus the grand total is 40 different ways.
Let me complete the problem...
How many total ways are there altogether? Well if we complete the ananlysis by looking at C(4,1)*C(6,2), we get
4*15 = 60 one woman committees and
C(6,3) = 20 no woman committees,
so there are a grand total of 120 possible committees, and thus
P(at least two women) = 40/120 = 1/3

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Assuming fair six-sided dice, we multiply the individual probabilities because we are asked to find the AND case...
so we have a 3 out of 6 for an odd on the red and
we have a 2 out of 6 for a 1 or a 2 on the blue...
multiplying we get
3/6 * 2/6 = 6/36 =
1/6

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8 out of every 13 tries...in the long run...

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C(12,10) or 12C10 is expressed as
(12!) / [(12-10)!(10!)] =
(12*11) / (2*1) =
66

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We say that is "ten chosen four at a time," and can be expressed as 10C4, or
(10!) / [(10-4)!(4!)] =
(10*9*8*7) / (4*3*2*1) =
210