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                   3x² - 6x - 2 + 8x + 14

                   3x² - 2x + 12

                   3(x² - 2/3x + ...) + 12                       (factor out the 3)

                   3(x² - 2/3x + 1/9) + 12 - 3(1/9)        (complete the square; add it in, subtract it out)

                   3(x - 1/3)² + 35/3

a² - b² + 3a + b

Factor:  (a + b)(a - b) + 3a + b

Sincea + b = -1:  (-1)(a - b) + 3a + b

                --->            -a + b + 3a + b

                                       2a + 2b

                                       2(a + b)

                                       2( -1)

                                         -2

Let  R  =  the rate of the plane.

With the wind, the rate will be  R + 50.

Against the wind, the rate will be  R - 50.

The total flight time will be  11 hours.

Let    T  =  the time going (with the wind)

  11 - T  =  the time returning (against the wind.

Since Distance = Rate x Time:

Going:              (R + 50)(T)  =  3000     --->     T  =  3000 / (R + 50)

Returning:  (R - 50)(11 - T)  =  3000     --->     11 - T  =  3000 / (R - 50)   

Combining:     11 - T  =  3000 / (R - 50)     --->     11 - 3000 / (R + 50)  =  3000 / (R - 50) 

--->     11  =  3000 / (R + 50) + 3000 / (R - 50) 

Multiply both sides the the common denominator  [  (R + 50)(R - 50)  ].

This will give you a quadratic; since it has a "nice" answer, either factor or use the quadratic formula.

Hi Melody,

They collected 192 dollars

Note that   sqrt 8 = 2 sqrt 2        sqrt 32 = 4 sqrt 2

 so you have

    (1 + 3/2 + 5/4 )  * 1/sqrt2

= 3 3/4 ( 1/ sqrt2)

= 15 sqrt 2 /8    

(x+6)^2 = x^2 + 12x + 36     means k = 22

(x-6)^2 = x^2 -12x + 36        means k = -2     Add 'em together........

Cannot you just type it in the calculator?

Thanks Tiggsy

I can follow all that.  Similar to mine just better :))

\(\displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}} =\frac{1}{4}\left\{\frac{1}{\sin^{2}(\pi/14)} +\frac{1}{\sin^{2}(3\pi/14)}+\dots +\frac{1}{\sin^{2}(13\pi/14)}\right\}.\\~\\ \displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}} =\frac{1}{4}\left\{\frac{2}{\sin^{2}(\pi/14)} +\frac{2}{\sin^{2}(3\pi/14)}+\frac{2}{\sin^{2}(5\pi/14)}+\frac{1}{\sin^{2}(\pi/2)} \right\}.\\~\\ \displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}} =\frac{1}{4}\left\{\frac{2}{\sin^{2}(\pi/14)} +\frac{2}{\sin^{2}(3\pi/14)}+\frac{2}{\sin^{2}(5\pi/14)}+1 \right\}.\\~\\ \displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}} =\frac{1}{4}+\frac{1}{2}\left\{\frac{1}{\sin^{2}(\pi/14)} +\frac{1}{\sin^{2}(3\pi/14)}+\frac{1}{\sin^{2}(5\pi/14)} \right\}.\\~\\ \displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}} =\frac{1}{4}+\frac{1}{2}\left\{24 \right\}.\\~\\ \displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}}=12.25 \)

LaTex

\displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}}
=\frac{1}{4}\left\{\frac{1}{\sin^{2}(\pi/14)} +\frac{1}{\sin^{2}(3\pi/14)}+\dots +\frac{1}{\sin^{2}(13\pi/14)}\right\}.\\~\\
\displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}}
=\frac{1}{4}\left\{\frac{2}{\sin^{2}(\pi/14)} +\frac{2}{\sin^{2}(3\pi/14)}+\frac{2}{\sin^{2}(5\pi/14)}+\frac{1}{\sin^{2}(\pi/2)}
\right\}.\\~\\
\displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}}
=\frac{1}{4}\left\{\frac{2}{\sin^{2}(\pi/14)} +\frac{2}{\sin^{2}(3\pi/14)}+\frac{2}{\sin^{2}(5\pi/14)}+1
\right\}.\\~\\
\displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}}
=\frac{1}{4}+\frac{1}{2}\left\{\frac{1}{\sin^{2}(\pi/14)} +\frac{1}{\sin^{2}(3\pi/14)}+\frac{1}{\sin^{2}(5\pi/14)}
\right\}.\\~\\

\displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}}
=\frac{1}{4}+\frac{1}{2}\left\{24
\right\}.\\~\\

\displaystyle \sum_{z}\frac{1}{\mid 1-z\mid^{2}}=12.25

Find the value of that root.

Hello Guest!

\(ax^2+bx+c\)

In the searched quadratic must apply:   \(b^2-4ac=0 \)

\(4x^2 + 16x + 8\ |\ b^2-4ac\neq 0\\ x^2 + 4x + 5\ |\ b^2-4ac\neq 0\\ 9x^2 - 6x + 1\ |\ \color{blue}b^2-4ac=(-6)^2-4\cdot 9\cdot 1=0\\ 2x^2 - 8x + 4\ |\ b^2-4ac\neq 0\\ 225x^2 - 30x + 9\ |\ \color{red}b^2-4ac< 0 \)  

The searched quadratic function is:

\(\color{blue}9x^2 - 6x + 1=0\\ x = {6 \pm \sqrt{36-4\cdot 9\cdot 1} \over 2\cdot 9}\\The\ value\ is\\ \color{blue}x=\dfrac{1}{3}\)

  !

A to Q ==17 letters

17 C 3 ==680 triplets.

The full list should be:

(961, 1147, 1271, 1333, 1369, 1457, 1517, 1591, 1643, 1681, 1739, 1763, 1829, 1849, 1891, 1927, 1961, 2021, 2077, 2173, 2183, 2201, 2209, 2257, 2263, 2279, 2419, 2449, 2479, 2491, 2501, 2537, 2573, 2623, 2627, 2701, 2747, 2759, 2773, 2809, 2867, 2881, 2911, 2923, 2993, 3007, 3053, 3071, 3127, 3131, 3139, 3149, 3193, 3233, 3239, 3293, 3317, 3337, 3379, 3397, 3403, 3431, 3481, 3503, 3551, 3569, 3589, 3599, 3649, 3713, 3721, 3737, 3763, 3811, 3827, 3869, 3901, 3937, 3953, 3959, 3977, 4033, 4061, 4087, 4141, 4171, 4181, 4183, 4187, 4189, 4223, 4247, 4307, 4309, 4331, 4343, 4387, 4399, 4429, 4453, 4469, 4489, 4559, 4601, 4633, 4661, 4687, 4699, 4717, 4747, 4757, 4819, 4841, 4847, 4859, 4891, 4897, 5029, 5041, 5063, 5069, 5123, 5141, 5143, 5183, 5207, 5251, 5293, 5311, 5329, 5353, 5371, 5429, 5459, 5461, 5561, 5609, 5617, 5633, 5671, 5699, 5723, 5767, 5777, 5891, 5893, 5917, 5959, 5963, 5969, 5977, 5989, 6059, 6077, 6157, 6161, 6241, 6283, 6313, 6319, 6431, 6439, 6497, 6499, 6527, 6533, 6557, 6649, 6667, 6731, 6767, 6887, 6889, 6893, 6901, 6943, 7031, 7081, 7169, 7171, 7261, 7303, 7313, 7367, 7373, 7387, 7493, 7519, 7571, 7597, 7663, 7729, 7739, 7747, 7811, 7921, 7957, 7979, 7991, 8023, 8051, 8083, 8137, 8201, 8249, 8357, 8383, 8453, 8479, 8509, 8549, 8611, 8633, 8777, 8881, 8927, 8989, 9017, 9047, 9167, 9179, 9271, 9301, 9313, 9379, 9409, 9523, 9563, 9701, 9727, 9797, 9869, 9991)>>Total==233

The sum of all possible values of BC is 12.

PQ = 22*sqrt(2)/3.

hint, try it with base10 

The smallest possible value of d is 7.

Do you know what y=cos(x) looks like?

Graph it on desmos if you do not know..     

What is its period?

Where does it first cross the x axis?

Angle XCD = 15 degrees.

p + q + r + s = 47.

The value of c is 10/3.

There are 1020 possible seatings.

The sum of all possible vaues of A is 482.

x/y = 2/7.

The area of the region is 27/2.

By the quadratic formula, v = -12.