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Sqrt[12, 296 / 3] =~64 - the average of the 3 numbers.

62^2 + 64^2 + 66^2 =12,296

[62 x 64 x 66] / 8 =32,736

Excellent, MG  !!!

>>>>>>>Sorry... I didn't see "even".....look at the guest's answer.....!!!

Look  at the grid below

7    8      9

4    5     6

1    2     3

First.....note that only  the corner points  and the middlle point can be reached in 4  moves from 1....so points 2, 4 , 6 and 8 cannot be reached in any set of 4 moves from 1

To reach 1 from 1 we have the following sets of moves

(1, 2, 1, 2, 1)  ( 1, 2, 5, 4, 1) (1, 4, 5, 2, 1) (1, 2, 3, 2, 1) (1, 4, 1, 4, 1) (1, 4, 7 , 4, 1)

To reach 3 from 1, we have these sets of moves

(1, 2, 3, 2, 3) ( 1, 2, 1, 2, 3 ) (1, 2, 3, 6, 3) (1, 2, 5, 6, 3) (1, 2, 3, 2, 1) ( 1, 4, 5, 6, 3) 

To reach 7 from 1 we have the same number of sets of moves as that of reaching 3 from 1

To reach the middle point 5 from 1 we have these sets of moves

(1, 2, 3, 6 , 5) (1, 2, 1, 2 , 5) (1, 2, 3, 2, 5) ( 1, 2, 5, 4 , 5) (1, 2, 5, 8, 5) (1, 2, 5, 2, 5) (1, 2, 5, 6, 5)

(1, 4, 7, 4, 5) ( 1, 4, 7, 8, 5) (1, 4, 5, 6, 5) (1, 4, 5, 8, 5) ( 1, 4, 1, 4 , 5)

To reach 9 from 1 we have

(1, 2, 3, 6, 9) (1, 2, 5, 6, 9) ( 1, 2, 5, 8, 9) (1, 4, 5, 6 , 9) (1, 4, 5, 8, 9) (1, 4, 7 , 8 , 9)

So.....we have a total of  4(6) + 12  =  36 possible moves

And 6 of these reach 9 from 1   -  B from A -.....so....the probability is  6 / 36   =  1 / 6

So,

Area= 180ft

Perimeter= 54ft

Simply meaning, two numbers that multiply to give 180 and add (x2 due to 4 sides) to give 54.

EG

\(\\X ⋅ Y\\ \\X+X+Y+Y\\ \)

Factors of 180:

1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180

Numbers have to be below 54 itself to add up to it

1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,

Find two factors that multiply to 180 that add to 54 (all 4 sides)

12 and 15

12*15= 180

12+12+15+15=54

Length must be 15ft

By diagonal, I assume you mean a diagonal line intersecting through the rectangle (unless you only mean the length of the patio itself).

\(\\12^2+15^2=369\\ \\\sqrt{369}=19.2ft\\\)

When squared that would be 369ft.

x = 8

 |6x^2-47x+15|  = 23 - which is a prime

Let the number of pieces in each pack =p
The total number of packs =50/p

2/3 - p/30 = 5p / 50, solve for p
p =5 - number of pieces in each pack
20/5 =4 packs of Cherry gum
30/5 =6 packs of Grapes gum
50/5 = 10 packs in total.

Check:
(20 - 5) / 30 =(5 x 5) /50
15/30 =25/50
1/2 = 1/2

You can add them up by grouping keenly: 3+6+9+2+5+8+1+4+7+0+3+6+9+2+5+8, pair 3 with the 7, 6 with the 4, 9 with the 1 and so on. Finally, you'll get \(\boxed{78}.\)

I'm fine, thank you!

We can set up an equation: \((33-x)+(24-x)+x+8=50\) . Solving, we get \(57-x+8=50, 57-x=42\)\(\) , so   \(\boxed{x=15}.\)

Well, PM....just thank the Lord that you're not that tree in your backyard....!!!

Good job, PM...!!!!

Where I live, it usually never rains, but today and yesterday, it rained a ton (of gallons). The wind is also blowing, so the tree in my backyard is losing a lot of leaves. Now my gutter is partially clogged... But other than that, I am fine. 

I am pretty sure you factored \(c^2 - 2c + 1 = 0\) incorrectly. 

Good, synth...hope you are OK, too....!!!!!

If the polynomial has 1 real root, then the discriminant mut equal 0. We can rewrite the polynomial as \(x^2 + cx + x + c\).

The discriminant is always \(b^2-4ac\), and setting a = 1 and b = c+1, we have \((c+1)^2 - 4c = 0\). Simplifying, we have \(c^2 + 2c +1 - 4c = 0 \Rightarrow c^2 - 2c +1 = 0\). Factoring, we have \((c-1)^2 = 0\). Taking the square root of both sides, we have \(c -1 = 0 \Rightarrow \boxed{c = 1}\).

There is only one solution since there is only once real root. The product of all possible values of c is 1. 

Hope this helps,

- PM

\(\text{1 real root indicates that the Discriminate }D = b^2 - 4ac \text{ is equal to }0\\ D=(c+1)^2 - 4(1)(c) = \\ c^2 + 2c + 1 - 4c =\\ c^2 - 2c+1 = 0\)

\((c-1)^2 = 0\\ c=1\)

Welcome, EP.....

[ Also note that -12 / 8 = - 3 /2 ]

For  f( f(x) to = 5

We have several possibilities

First

(x + 3) + 3 = 5

x + 6 = 5

x = -1

But.....note that  f(f(-1)) = f (-3)  = 5   so....-1  works

Also

(x^2 - 4)^2 - 4 = 5

x^4 - 8x^2 + 16 - 9 = 0

x^4 - 8x^2 + 7 = 0        factor

(x^2 - 7) ( x ^2 - 1)

We have 4 possible solutions

x = -√7, √7, -1 , 1

 x = - √7  will produce  because f(f(-√7)) = f (3) = 6

We have seen that x = - 1 works

x = 1 will work because  f(f(1)) = f(-3) = 5

x = √7  will work because   f(f(√7)) =  f(3)  =  5

Another possiibility is  that

(x +3)^2 - 4 = 5

x^2 + 6x +  9 - 9 = 0

x^2 + 6x = 0

x(x + 6) = 0

x =  0   or x = -6

f(f(0)) = f(-4) =  12

f(f(-6)) = f(-3) = 5

The last possibility is that

(x^2 - 4) + 3 = 5

x^2 - 6 = 0

x = -√6  or x = √6

f(f(-√6)) = f (2) = 0

f(f(√6)) = f(2) = 0

So....the x values  where f(f(x) ) = 5  are

x = -1, 1, -√7, √7, -6

From 1 to 30, there are 10 multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30. The sum of the units digits is 3+6+9+2+5+8+1+4+7+0 = 45. 

Since the digits go up to 50, not 60, we stop the next list of addents up to 18. 3+6+9+2+5+8 = 33. 33+45 = \(\boxed{78}\).

You could also just use brute force and add them all up.