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Nice keep it up man I like what you did So i sighned up. Maby we could be friends.

Colored  ones  =  Total  - White ones  =   8  - 5  =  3

So

Colored Socks / Total     =    3 / 8   =  .375  =     37.5 %

x ( x + 2) (4x  - 1)

Taking the first two terms  we have   x ( x + 2)  =  x^2 + 2x

So now, we are multiplying

(x^2 + 2x)  ( 4x  - 1)       

Distribute the terms in the first set of parentheses over (4x -1)

x^2  (4x - 1)      =               4x^3      -  1x^2

+

2x(4x  - 1)      =                               +  8x^2    -  2x

__________________________________________

                                        4x^3     +   7x^2    -   2x

Since there are  180°  in every triangle,   m∠J  =  180° - 73° - 52°   =   55°

Now we can use the law of sines to find the length of  HK .

\(\frac{HK}{\sin55°}\,=\,\frac{14}{\sin52°}\)

                                    Multiply both sides of the equation by  sin 55°

\(HK\,=\,\frac{14\sin55°}{\sin52°}\)

HK   ≈   14.6     yards

x^2   +  ax  +  5a  = 0

Let m, n  be the integer roots...this implies that....

(x - m) (x - n)  =  0

x^2  - (m + n)x  + mn = 0

So....equating terms.....we have this

-m - n  = a   ⇒  -5m -5n   =  5a

mn  =  5a

So this implies that

-5m - 5n  =  mn

-5n - mn  =  5m

-n ( 5 + m)  =  5m

n  =   -5m / ( m + 5)

Note that, as m → ± inf,   n →  -5

Also.....m   must be  a multiple of 5  since  -5m  is a multiple of 5

So  we can  test some values

m       n         -m - n  =   a

0        0                          0

20     -4                       -16

All positive values of m > 20  means that -5 < n < -4   so no other integer values are possible for in this direction

-10   -10                       20

-30   -6                         36

All negative values of m < -30  means that  -6 < n < -5 so no other integer values are possible for n at all

So the possible values for "a" are :

  -16, 0, 20  and 36

Number of problems normally answered per day   is

p * t

So......when he drinks coffee, he solves

(3p + 7)  (t - 4)  problems....and this is twice the number he normally answers...so...

(3p + 7) (t - 4)    =   2p*t

3pt  +  7t   -  12p  -  28  =  2pt

pt  +  7t  -  12p  -  28  =  0

We have an equation with two unknowns....so....there will possibly be more than one solution

WolframAlpha shows that the (possible) feasible solutions are

p = 1     t  = 5   =    5 problems on a regular day

p = 7     t  = 8  =    56 problems on a normal day

So as a check

(3*1 + 7)(5 - 4)  = (2)(1*5)  ??

(10)(1)  = 10  is true

(3*7 + 7) (8 - 4)  = (2)(7*8)  ???

(28) (4)  = 112

112 =  112    is also true

So.......the number of problems that he answers when he drinks coffee is either

(3)(1) + 7    =  10  problems per day

or

(3)(7) + 7  =   112 problems per day

Meyer rolls two fair, ordinary dice with the numbers 1,2,3,4,5,6 on their sides. What is the probability that AT LEAST one of the dice shows a square number?

So one of the numbers must be 1 or 4

the posibilities are 

P(1,anything)  = (1/6)*1 =   1/6

P(4,anything) = 1/6

so

P(1, or 4 and any number on the other die) = 2/6

P(any number, 1 or 4) = 2/6

so together that is 4/6  trouble is some combinations have been added twice.

P(first die is 1 or 4, AND second die is 1 or 4) = 2/6 * 2/6 = 4/36 

So the prob that you want will  be  2/6  +  2/6  - 4/36 = 20/36 = 5/9

You can check it by counting them.  To get all the combinations just do it in a 6 by 6 grid.

Mmm Lets see.

\(f(x)=\sqrt[2]{x+1}-2\)

this is just

\(f(x)=\sqrt[2]{x+1}\)      dropped 2 units

so I will look at this one:

\(y=\sqrt[2]{x+1}\\ \qquad \text{I first note that }y\ge0\qquad and \qquad x\ge-1\\ y^2=x+1\\ x=y^2-1\\ \text{This is a sideways parabola. It will open up on the positive side }.\\ \text{The parent function for this one is }x=y^2\\ \text{The -1 means that it is moved 1 unit to the left }\)

SO what do we have - we have to go backwards now:

Graph  \(x=y^2\)  which is a sideways parabola with vertex (0,0)

Now move it 1 unit to the left.

Now only include the bits where \( x\ge-1 \quad  and  \quad y\ge0\)

NOTE: This is now the same as  \(y=\sqrt{x+1}\)

Now drop it two units

and then you should have it.       \(y=\sqrt{x+1}-2\)

I will show you how this graph grows from the parent (or is that grandparent) function:

Here is how the graph is built up.

You can play withh this, if you press the coloured circles on the left your graph will diappear or reappear, so you can see how I built the graph :)

The questions second half is getting cut off for a unknown reason.

You are correct, EP !!!!

Just give us the address of the old question please.

NOT the address of the screen capture!

Ok....let's pick some points  and put them into the function to get y.....

To save time.... i'm going to pick some points for x  and have WolframAlpha  generate the associated y values

You can plot these for yourself to get a rough idea of the graph 

sqrt(x + 1) - 2 where

x = {-1, -0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3}

y  =  {-2, -1.29289, -1, -0.775255, sqrt(2) - 2, -0.418861, sqrt(3) - 2, -0.129171, 0}

The mathematician solves 112 problems that day.

We can rewrite the equation as (3p+7)(t-4) = 2tp, then we can solve for t with 4*(3-(14/p+7)).  The expression (14/p+7) can only be 1, which results in a value of 7 for p.  We can then solve the problem easily: (3*7 + 7)(8-4) = 28*4 = 112.

Thanks so much! I don't know how I missed that :I

I was wondering how you would graph this by hand

Area of semi-circle  =  1/2 x pi r^2 = 1/2 (pi)(3)^2 = 14.137

Area of triangle = 1/2 b x h = 1/2 x 6 x 5 = 15

total area = 14.137 + 15 = 29.137 sq units

This is the square root graph   shifted to the  left by 1 unit and down 2 units

See  both here : 

https://www.desmos.com/calculator/c2df9wir6j

NOT sure, but I think the little marks on the side of the triangle mean that both sides are '12' each.

Pythagorean theorem:

x = sqrt (12^2  + 12^2) = sqrt (288) = 16.97 units

5 x  sqrt ((1.33-4)^2 + (1-3)^2) = 16.68 miles

We have the opp and hyp known   .....so....use the sin inverse 

asin ( 66/ 79 )   =  x  ≈  56.66°

We  can use the tangent to sollve this

tan (62)  =  x  / 40      multiply both sides by 40

40 tan (62)  =  x  ≈  75.23   yd

We can use the Pythagorean Theorem to answer this

The hypotenuse  =  84     and one of the legs  = 50

So....the distance from the base of the ramp to the base of the wall is

√ [  84^2  -  50^2  ]   =  √ 4556  in   ≈    67.5  in 

Thanks Chris,

At least I hope I have found my problem :))

180 dgrees - 33 degrees - 104 degrees = 43 degrees = angle g   (and m)

(f × g)(x)  =  ( 1 / (log x) )( 1 / (8 + x) )

Since  log x  is part of the function, we have the restriction  x > 0

Since  log x  is in a denominator, we have the restriction  log x  ≠  0

log x  ≠  0

x  ≠  10⁰

x  ≠  1

Since  8 + x  is in a denominator, we have the restriction  8 + x  ≠  0

8 + x  ≠  0

x  ≠  -8               Notice that this is already accounted for with  x > 0

So the domain is all real  x  values such that  x > 0   and   x  ≠  1