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\(\frac{1}{\frac{1(1+1)}{2}}+\frac{1}{\frac{2(2+1)}{2}}+\dots + \frac{1}{\frac{2018(2018+1)}{2}}\)

\(\frac{2}{1(2)}+\frac{2}{2(3)}+ \dots+\frac{2}{2018(2019)}\)

\(2(\frac{1}{1(2)}+ \frac{1}{2(3)} + \dots + \frac{1}{2018(2019)})\)
\(\text{Apply } \frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}\)

\(2(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3} +\dots -\frac{1}{2019})\)

\(2(1-\frac{1}{2019})\)

\(\frac{4036}{2019}\)

Hope someone will help.

(xy) / (x + y)  = 1

(xz) /(x + z) =1

(yz) ( y + z)  = 3

xy = x + y  

xz = x + z   →   xz - x = z  →  x ( z - 1)  = z →  x =  z / (z -1)

yz =  3(y + z)  →   yz = 3y + 3z →  yz - 3y = 3z →  y ( z - 3)  =  3z →  y = 3z / ( z -3)

xy  = x + y

[ z / ( z -1) ] [3z / ( z -3) ] = z / (z - 1) + 3z / ( z -3)

[ 3z^2 ] / [ (z -1) ( z - 3) ] =  z (z -3) + 3z ( z -1) / [ (z -3) (z -1) ]

3z^2  = z ( z -3) +3z ( z -1)

3z^2  = z^2 - 3z + 3z^2 - 3z

z^2 - 6z  =  0

z ( z - 6 ]  = 0

z = 0   (reject)

z =  6

y = 6

x  = 6/5

To find the height of the box, use the given relations in the problem and the formula for the total surface area of a rectangular box, work through algebraic simplifications and potentially systems of equations to solve for the dimensions, which would give the height.

Let x be the number of dollars my brother has. Since we will have the same amount of money if I give him 5 dollars, I must have x + 10 dollars. (Make sure you see why I don't have x + 5 dollars. If I give him 5 dollars of my x + 10 dollars, then we will both have x + 5 dollars.)

If he gives me 17 dollars, then he will have x - 17 dollars and I will have x + 10 + 17 = x + 27 dollars. I'll then have twice as much money as he has, so

x + 27 = 2(x - 17) Expanding the right-hand side gives x + 27 = 2x - 34, so x = 61. So, my brother has $61, and I have $71

Where is the diagram?

3!*4! = 6*24= 144 ways

$3.05

I still gave you a point  for a good  effort !!!!

\(\sqrt{1-\sqrt{2+\sqrt{3+\sqrt{x}}}} \)

Note   that  even if x =  0   (as small as it  can  be),,,we have

1 - sqrt [ 2 + sqrt (3) ]   which is less than   0......so the quantity under the radical isn't real

If x >  0     then    sqrt [ 2 + sqrt (3 + sqrt (x)) ]    will still be >1  which makes the quantity under the radical still < 0] 

So.....this  can't have a domain of real numbers

Srry mb gang 

Assuming "a" not equal to  0, multiply through by a

a^2/3 + a  = (a + 3) - (a^2 + 2)

(1/3)a^2  + a  = -a^2 + a + 1          simplify as

(4/3)a^2  =  1             multiply both sides by 3/4

a^2  = 3/4                  take both roots

a =  sqrt (3)  / 2      or     a =  -sqrt (3)  / 2

a/3 + 1 = (( a+3) - (a^2 + 2))/a

a/3 + 1 =( a^2 - a - 1 ) / a 

a ( a/3 + 1 ) = a^2 - a - 1 

a^2/3 + a = a^2 - a -1 

a^2/3 = a^2 - 2a -1 

0 = (2a^2 - 2a) / 3  - 1 

then we can plug this equation into the quadratic equation, 

this gives us the answer 

(3+sqrt 15 )/2 or (3-sqrt15)/2 

please if anyone could check my answer that would be a fire 

5N / ( N + 2)  =  6

5N  = 6(N + 2)

5N  = 6N + 12

-N =  12

N =   -12

L = H  - 5

H = L + 5

L = 2W

W = L/2

W = ( H - 5) / 2

SA = 2 [ LW + HL + HW ]

234  = 2  [  (H - 5)(H - 5)/2 + H(H  -5)  +  H (H - 5)  / 2 ]

234  = (H - 5)(H-5)  + 2H (H -5)  + H(H -5)

234 =  H^2 - 10H + 25 + 2H^2 - 10H + H^2 - 5H

4H^2  - 25H  + 25  = 234

4H^2  - 25H - 209 =  0

H  =   [25 + sqrt [ 625 -  4*4 * -209 ] ] / 8   =  [25 + sqrt [ 3969 ] / 8  = [25 + 63 ] / 8  = 88 / 8  =  11

\(\frac{(2x + 3)(x - 7)}{\sqrt{x}} \)

The numerator is  defined for  all x

The  denominator   cannnot  = 0   and  x  must be > 0

So.....the most restrictive interval  of the domain  is

 (0, inf )

Take the derivative with respect to x  and  set to  0

2x + 12  =  0

x  =  -6

Take the derivative with respect to y and set to 0

2y - 16 = 0 

y = 8

Minimum  is   (-6)^2 + (8)^2 + 12 (-6) -16(8) + 8   =   -92

Bosco is correct......no solution.....

M - 5 = B + 5  → M = B + 10    (1)

M + 10  = 2 ( B - 10)

M + 10  = 2B - 20

M  = 2B - 30     (2)

Set (1)  = (2)

B + 10  = 2B  -30

40 = B

Your brother has $40     

Check

Your brother has 40 , you  have 50

When you give him 5, he has 45, you have 45

Instead, when he gives you 10 , he has 30  and  you have 60

Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to $7$ times the largest of the seven integers. What is the smallest integer that Pearl wrote down?   

Call the smallest integer x.    

  (x) + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6)  =  7 • (x+6)       

                                                                   7x + 21  =  7x + 42   

I don't see how you can get a solution to the problem as stated.   

_.

First, simplify the function to: \(x=2y^2-13y-1\)

Setting \(x'=0\), \(4y-13=0\) so \(y=\dfrac{13}{4}\). Your next step is to find \(x\). Then \((x,y)\) is the vertex.

Find the vertex of the graph of the equation y = -2x^2 + 8x + 15 - 3x^2 + 7x - 25   

Combine like terms                                        y  =  –5x² + 15x – 10    

Take the first derivitive                                   y'  =  –10x + 15   

The first derivitive defines the slope of the   

curve at any point.  When the derivitive is   

equal to zero, that's where the parabola   

turns around and goes the other way.   

Set equal to zero, solve for x                          –10x + 15  =  0   

                                                                             x  =  –15 / –10  =  3 / 2  

This is the x-coordinate of the vertex    

Plug it in to the original equation                    y  =  –5x² + 15x – 10    

This is to see what y is when x=3/2    

                                                                       y  =  (–5)(3/2)² + (15)(3/2) – 10   

                                                                       y  =  (–5)(9/4) + (15)(3/2) – 10    

                                                                                –45 + 90 – 40        

Find the common denominator                       y  =  –––––––––––––  =  5 / 4         

                                                                                          4        

This is the y-coordinate of the vertex    

Put them together and the vertex is at (3/2 , 5/4)   or, if you prefer, (1.50 , 1.25)    

I think there's a formula for finding the vertex without using the first derivitive   

but I don't know what it is.  Maybe somebody else will chime in.   

_.

The given equation is:
2x2+4x−1=x2−8x+3

Rearranging terms, we get:
2x2+4x−1−x2+8x−3=0
x2+12x−4=0
Now, the sum of the squares of the roots of this equation can be found using the formula:

Sum of the squares of the roots=(Sum of the roots)2−2(Product of the roots)
 

For the equation $x^2 + bx + c = 0$, the sum of the roots is $-b$ and the product of the roots is $c$. Therefore, for our equation $x^2 + 12x - 4 = 0$, the sum of the roots is $-12$ and the product of the roots is $-4$.

Plugging these values into the formula, we get: Sum of the squares of the roots=(−(−12*12))−2(−4)
=(12*12)+8
=144+8
=152

So, the sum of the squares of the roots of the given equation is 152.