Tetration

The correct answer is: -9x²+8x+7   (checked by wolfram alpha)

I think the wrong answer originated from here:

 "... 2x + 8 - (1- 3x)^2

now here we will use an identity for the expression in brackets 

Identity is (a-b)^2= a^2 - 2ab + b^2 

now we will use this identity to solve the bracketed expression!

= 2x + 8 - 1 - 2x1x3x + (3x)^2 ..."

I'll solve it like this:

2x + 8 - (1 - 3x)² =

2x + 8 - (1 - 6x + 9x²) =

2x + 8 - 1 + 6x - 9x² 

So it's mistakes made based on the minus before the paranthesis. 

This is not an equation, and x can be any number. 

1/2(-p+23+-p+15)=6p-23

 You mean this, right?:

(1/2)[-p + 23 + (-p) + 15] = 6p - 23     calculate: -p - p, and 23 + 15:

(1/2)(38 - 2p) = 6p - 23                 Find the product on left side:

19 - p  = 6p - 23                      Add p + 23 on both sides:

42 = 7p                      Divide both sides by 7

p = 6

(x-2)/(x+4)-5/6=0     Add 5/6 to both sides:

(x - 2)/(x + 4) = 5/6      Multiply both sided by x + 4:

x - 2 = 5(x + 4)/6          

x - 2 = (5x + 20)/6 = 5x/6 + 10/3      

x - 5x/6 = 10/3 +2                Get common denominator (bottom) by x = 1x/1 = 6x/6:

6x/6 - 5x/6 = 10/3 + 6/3       

x/6 = 16/3                            Multiply both sides by 6 = 2*3:

x = (16*2*3)/3 = 16*2 = 32

Double factorial (aka semifactorial) is a mathematical operation in which one multiplies "every other" natural number up to a given number. A more precise definition is:

$$n!!=\prod \limits_{i=0}^{k}(n-2i)= n(n-2)(n-4)(n-6)...$$

Where k = (n/2) - 1 

Or read more on the "Double Factorial" Wikipedia web page :)

20 goes one into 25 one single time, and leave a remainder of 5. 

25 mod 20 = 5

Where "mod" stands for modulo. 

Pi ≈ 3.141592

That's the digits I remember by heart :)

When I think about it, the next digit is 6, so it should have been written:

Pi ≈ 3.14159265

Eureka!

4!!+(4+4)/4

Where "!!" stands for double factorial.

$$1\frac{1}{4}*4\frac{1}{2}=$$

$$(1+\frac{1}{4})*(4+\frac{1}{2})=$$

Left, outer, inner, right:

$$4+\frac{1}{2}+1+\frac{1}{8}$$

Multiply(*) both the numerator (top) and the denominator (bottom) in fraction nr. 1 by 4 to get a common denominator:

$$4+\frac{4}{8}+1+\frac{1}{8}$$

$$5+\frac{5}{8}$$

Remember that: $$a\frac{b}{c}= \frac{ac +b}{c}$$, we get:

$$5\frac{5}{8}=\frac{40+5}{8}=\frac{45}{8}=5.625$$