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No prob  !!!

Thank you! 

sin^2x - 2sin x + 1  = 0        factor as

(sin  x - 1) (sin x - 1) = 0

Either factor will give the same solution

sin x - 1  =   0

sin x = 1

arcsin (1)  = x  =  pi/ 2

See the graph here : https://www.desmos.com/calculator/umo72uhxdo

The intersections (solutions) occur at x = -3   and x = 1

This is even....

To see why.....if  f(x) = f(-x)     it is even

So

f(x)  =  -2x^4 + 3x^2

And

f(-x)  =  -2(-x)^4  + 3(-x) ^2  =    -2x^4 + 3x^2  =   f(x)

The solutions to this are not integers ....see here :

https://www.wolframalpha.com/input/?i=x%5E2+-+3+%3D+1+-2x

(f + g) (x)  =   f(x )  + g(x)  =

[8x^2 + 16x + 6 ] + [ x^3 - 3x^2 - 9 ]  =

x^3 + 5x^2 + 16x - 3 

Yep, this is known as the "Partial Fraction Decomposition." 

You can try guessing and checking by first placing all the large boxes, then slowly decreasing the number.

But we can't say 12 large boxes and 0 small boxes, because there has to be an amount for the small boxes.

Hint: To find the number of small boxes, it has to be a multiple of 10. So, what number must 8 be multiplied, that when the number is subtracted from 96, it is divisible by 10?

2.

( u + 2v) (dot) w  =

u (dot) w   +  2v (dot) w  =

4   + 2(v (dot) w)  =

4 + 2( 5)  =  10

(w - u) (dot ) v =

w(dot)v   -  u(dot)v =

5  -  3     =    2

(3v - 2w)(dot) u =

3v (dot)u   -  2w (dot) u =

3 [ v (dot) u ]  - 2 [ w (dot) u ] =

3 [3]  - 2[4]  =

9  - 8  =   1

So

( u + 2v) (dot) w  ,  (w - u) (dot ) v , (3v - 2w)(dot) u  =   10, 2 , 1

f(f(f(x)) = - 3

Take the inverse of both sides

f^-1 f(f(f(x))  = f^-1 (-3)     =   - 4  and 0

f(f(x)  =  f^-1 (-3)  =  - 4  and 0

Take the inverse of both sides again

f^-1 f(f(x) )  = f^-1(-4)        and    f^-1 f(f(x))  = f^-1(0)

f(x)  = -2                     and      f(x) = -6           and      f(x)  =   2

Take the inverse one last time

f^-1f(x)  = f^-1(-2)         and  f^-1 f(x) = f^-1(-6)    and   f^-1 f(x)  = f^-1(2)

x =  f^-1(-2)       and   x  = f^-1(-6)     and    x  = f^-1 (2)

Since  x = - 2  is a line of symmetry

Then   f^-1(-2)  can be written as   -2 - a, -2 + a

And   f^-1(2)  =  can be written as    -2 - b, -2 + b

And f^-1(-6)   = not on graph

So....the sum of the coordinates =

(-2 - a) + (-2 + a) + (-2 - b) + (-2 + b)  =       -8

the answer is 3 since the first one is one color, the second one will be the opposite color and the third one can be any color.

Here is another way of doing it:

1/7 =0.142857...........etc.

1- Multiply the above fraction by 16 and subtract the integer part:

16 x 0.142857......... = 2.2857142857142857142857142857143 - 2 =0.28571428..........etc. Keep this 2 and:

2 - Multiply the remaining fraction by 16:

0.2857142857...... x 16=4.571428571428571428571428571 - 4    =0.57142857.............etc. Keep this 4 and:

3- Multiply the remaining fraction by 16:

0.57142857......      x 16 =9.14285714285714285714285714 - 9     =0.142857142...........etc. Keep this 9 and:

4 - List them from top to bottom with a decimal in front:

0.249249249249........As you can see the decimal reverts to 1/7 =0.14285712..........etc. in (3) above.

        4x                              A                   B                   C

____________  =        _______    +     _____    +    _______       

(x - 5)(x - 3)^2               (x - 5)                (x - 3)          (x - 3)^2

 Multiply through by (x - 5) ( x -3)^2

4x   =    A(x - 3)^2   +  B(x- 5)(x - 3)  + C(x - 5)      simplify

4x  =  A(x^2 -  6x + 9)  + B(x^2 - 8x + 15)  + C(x - 5)

4x  = (A + B)x^2    + (-6A - 8B + C)x + (9A + 15B - 5C)

Equating coefficients, we have

A + B  =  0     ⇒   B = -A      (1)    

-6A - 8B + C  = 4    (2)

9A + 15B - 5C =  0     (3)

Subbing  (1)  into (2)  and (3)   we have

-6A + 8A + C   = 4   ⇒   2A + C  =  4   ⇒   6A + 3C = 12    (4)

9A - 15A - 5C  = 0  ⇒  -6A - 5C = 0        (5)     

Add (4) and (5)

-2C =  12

C = -6

And using (4) to find (4)

6A + 3(-6)  = 12

6A - 18 = 12

6A = 30

A = 5

And B = -A =    -5

So   (A,B,C)  = (5, -5, - 6)

For every vote that his opponent receives, Danny gets 5/3  votes

So.....the number of votes that Danny receives is 312(5/3)  = 520 votes

So    312 + 520   =   832 votes were cast

The day after the school election it was reported that Danny won the election for class president. For every three votes his opponent received, Danny received 5 votes. Every student voted exactly once. If the students voted only for either Danny or his opponent and Danny's opponent received 312 votes, how many students voted in the election? 

Danny received  =          5 of every 8 votes

Opponent received  =     3 of every 8 votes

Let V stand for the total number of votes cast.

The opponent's 3/8 of the votes was 312  so  (3/8) times V  = 312

(3/8)(V)  =  312

V  =  (312)/(3/8)              ( i.e., 312 divided by 3/8 )

To divide by a fraction, you invert the divisor and multiply

V  = (312) x (8/3)  =  2496/8  =  832 total votes 

.

To see this, imagine that  just 4 people were at the party

The first person shakes 3 hands

The second person shakes 2 hands

The third person shakes 1 hand

The last person has already shaken hands with everybody

So...the number of handshakes between 4 people =  (4)(3) / 2 =  6

So....it appears that the number of handshakes between n people  is just n(n - 1) /2

So.....we can solve this

n(n - 1) / 2 = 66      multiply both sides by 2

n(n - 1) = 132   simplify

n^2 - n = 132

n^2 - n - 132 = 0     factor

(n - 12)  ( n + 11) = 0

Setting each factor to 0  ans solving for n  ives n = -11  (reject)   or n = 12 (accept)

So....there wqere 12 people

Thanks

Daphne has a rope that is 60 meters long. She wants to use it to mark the boundary of a circle whose radius is an integer. What's the largest possible radius for her circle, in meters? 

Boundary is another way of saying circumference. 

Circumference = pi times diameter

60  =  3.14 x D 

D  =  60/3.14  =  19.11

Radius is half the diameter so the radius would be 9.55 

Truncating to an integer, the largest radius would be 9 meters 

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It is sequence A052918 on OEIS. a₃ is 5(26) + 5 = 135 as the transition formula can be rewritten as \(a_{n+1} = 5a_n + a_{n-1}\).

https://oeis.org/A052918 <-- Here's a link to the page about sequence on OEIS.

1. Third option

2

 \(f(-2) = 3-(-2)^2 = -1\\ g(-2) = -1\\ \therefore f(x) = g(x) \text{ when } x = -2.\)

\(\quad\left(\dfrac{f}{g}\right)(x)\\ =\dfrac{4x^2+6x}{2x^2+13x+15}\\ =\dfrac{2x(2x + 3)}{(2x + 3)(x + 5)}\\ =\dfrac{2x}{x+5}\)

"What are the first 10 digits after the decimal point (technically the hexadecimal point...) when the fraction 1/7 is written in base 16?"

\(x = \frac{1}{7}\\x_d=0.\overline{142857}\\x_h=0.\overline{249}\)

I'll leave you to work out the other 7 digits!!