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THX SO MUCH GUYS!!!!!!

For this, we need to find a quadratic where you have to get a factored equation like (x+h)^2=0.

For this, you should factor all of these quadratics to look for the equation/expression that will get you (x+h)^2=0.

We have \(\frac{x}{y^3}\) and \(x\sqrt{z}\) are the two restrictions...

\(3\sqrt{12}=y*\sqrt{75}, y=\frac{6}{5}=1.2.\)

Multiplying  a  polynomial  by a non-zero constant  does  not  change its degree

So    3* f(x)   still has degree  8

And  if  h(x)  is  degree 9,  then  g(x)   must  have  degree  9    because

Degree  8  + Degree 9    will  result  in a Degree 9  polynomial  = h(x)

Look at the other post about this I've seen before--

https://web2.0calc.com/questions/help_90704

Dividing is the same as multiplying the reciprocal of the number that is being divided. (Divisor)

3/5 divided by 1/3 is 3/5 times 3 which is 9/5 or 1 4/5.

Hope this helped! 

By Vieta....

We have  the form   Ax^2 + Bx + C  = 0

Let the roots  be  R₁ and R₂

The sum of these  =  -B/A  =   -4/3  =   R₁ + R₂     (1)

And the product of these =  C/A  =   =  12/3 =  4  = R₁*R₂  which implies that  8 = 2*R₁*R₂

Square  (1)  and we get that

R₁^2  + 2*R₁*R₂  + R₂^2  =  16/9

R₁^2  + 8  + R₂^2  =  16/9

R₁^2  + 72/9  + R₂^2  = 16/9      subtract 72/9  from both sides

R₁^2 + R₂^2  =  16/9 - 72/9  =  -56/9

We are asked to find: \(x^2+y^2\), for \(x\) and \(y\) are the roots of the equation.

\(3x^2+4x+12\) has a sum of \(\frac{-4}{3}.\)

\(3x^2+4x+12\) has a product of \(\frac{12}{3}=4.\)

Use the manipulation:\((x+y)^2-2(xy)=x^2+y^2\).

Thus, the answer is \((\frac{-4}{3})^2-2(4)=\frac{16}{9}-\frac{72}{9}=\frac{-56}{9}.\)

Thanks, Cal  !!!

Here's another way to solve this with linear programming

Set the equations up as equalities

2x + 5y  =10 ⇒    6x  + 15y  = 30      

3x + 4y  = 12 ⇒   -6x  -  8y   = -24          add these

7y = 6

y = 6/7

And   

2x  + 5(6/7)  = 10

2x  + 30/7 = 70/7

2x  = 40/7

x = 20/7

These  are  the  (x, y)  values  that will  maximize   8x + 13y

And  this  max  is    8(20/7) + 13(6/7)  =   [160 + 78 ] / 7  = 238/7   = 34

This can be  confirmed with this graph :    https://www.desmos.com/calculator/nwl5zh1adg

The max  will occur at  the "corner" point of  the intersection of the above inequalities ⇒  (20/7, 6/7)

3/5   /   1/3     is the same as        3/5  *   3/1  = 9/5

Thank you!! This really helped!!

Thx, Guest...I haven't heard of this before !!!!......I'm going to have to investigate "Ptolemy's Theorem".....

We have  OP + OQ  = 5 + 5   = 10

Also....the  curved part  is a 3/4 circle with a radius of 5....so....its perimeter  is   (3/4) (2pi) R  =

(3/4) (2pi) (5)   =   30 pi / 4  =   15 pi / 2   = 7/5 pi

Putiing this all together....the total perimeter is

10 + 7.5 pi ≈  33.56  units

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I simply wrote a simple brute force and ignorance program to find it!

-x would be the opposite of 1/3, and the absolute value of 7/4 would also be the opposite.

This gets you -1/3+7/4 which I think you can figure out yourself.

Hope this helped! 

a)Since there are 13 spades and 13 diamonds (26 total) (Spades and Diamonds)/(Total Amount) would be 26/52 which is 1/2.

b)There are 13 spades, 13 hearts, and 13 diamonds. (39 total) 39/52 is equal to 3/4. Therefore, 3/4 is the answer.

c)There are 4 aces and 13 hearts. However, one ace is a heart as well, making there be 16 hearts and aces total. 16/52 is equal to 4/13.

Hope this helped!

We can write the inequalities 2x + 5y ≥ 10 and 3x + 4y ≥ 12. To get the value of 8x + 13y, add the first inequality to twice the second inequality. This gives you 8x + 13y = (2x + 5y) + 2(3x + 4y) ≥ 10 + 2 · 12 = 34 . Hope this helped!

The volume of the cube would be 9.5*9.5*9.5 as stated below. Then, subtract the value of the ball (Sphere) inside, using the formula V=4/3(pi)r^3.

Hope this helped!

I am with my new account now 

Thank you Melody! I think I got it! I greatly appreciate all of you who helped me with this problem!

x(3)+2x(5)=1131

3x+10x=1131

13x=1131

x=87

2x=174

174 adults+ 87 students= $1131

87(3)+174(5)=$1131

261+870=$1131

$1131=$1131

cos x

reflected

   -cos x

amplitude 5

    -5 cos x

period 6

   -5 cos ( pix/3)

phase shift .25 r

   -5 cos (pi x/3 - pi/12)

The volume of the cube is  9.5 x 9.5 x 9.5  cu. in.

Find this value and subtact the volume of the ball.

Thanks for the help!