heureka

factor completely y^2 +yt-12t^2

$$\small{\text{$ \mathbf{ y^2 +yt-12t^2 = (y-3t)(y+4t) } $}}$$

1:4 =5:a       a = ?

$$\\ \small{\text{$ \begin{array}{rclcrcl} 1\cdot \frac{4}{1} &=& 4 &\rm{~~or~~}& 1\cdot \mathbf{4} &=& 4\\\\ && &\rm{~~so~~}&5\cdot \mathbf{4} &=& a\\\\ &&&&\textcolor[rgb]{1,0,0}{a}&\textcolor[rgb]{1,0,0}{=}& \textcolor[rgb]{1,0,0}{20} \end{array} $}}$$

12:8:4 = 39:A:B what is A and B

$$\\\small{\text{$ \begin{array}{rclcrcl} 12\cdot \frac{8}{12} &=& 8 &\rm{~~or~~}& 12\cdot \mathbf{\frac{2}{3}} &=& 8\\\\ && &\rm{~~so~~}& 39\cdot \mathbf{\frac{2}{3}} &=& A\\\\ &&&&\textcolor[rgb]{1,0,0}{A}&\textcolor[rgb]{1,0,0}{=}& \textcolor[rgb]{1,0,0}{26}\\\\ \hline &\\ 8\cdot \frac{4}{8} &=& 4 &\rm{~~or~~}& 8\cdot \mathbf{\frac{1}{2}} &=& 4\\\\ && &\rm{~~so~~}& A\cdot \mathbf{\frac{1}{2}} &=& B\\\\ &&&& 26\cdot \mathbf{\frac{1}{2}} &=& B\\\\ &&&&\textcolor[rgb]{1,0,0}{B}&\textcolor[rgb]{1,0,0}{=}& \textcolor[rgb]{1,0,0}{13} \end{array} $}}$$

Given vec f = 7 vec i + 4 vec j and vec q = 5 vec i -7 vec j, find vec f + vec q. vec f + vec q = vec i + vec j

$$\begin{array}{rcl} \vec{ \rm{f} } &=& 7\vec{ \rm{i} }+4\vec{ \rm{j} }\\ \vec{ \rm{q} } &=& 5\vec{ \rm{i} }-7\vec{ \rm{j} } &\\ \hline &\\ \vec{ \rm{f} } + \vec{ \rm{q} } &=& 7\vec{ \rm{i} }+4\vec{ \rm{j} } + 5\vec{ \rm{i} }-7\vec{ \rm{j} }\\ \vec{ \rm{f} } + \vec{ \rm{q} } &=& 7\vec{ \rm{i} }+5\vec{ \rm{i} } + 4\vec{ \rm{j} } - 7\vec{ \rm{j} } \\ \vec{ \rm{f} } + \vec{ \rm{q} } &=& (7+5)\vec{ \rm{i} } + (4-7)\vec{ \rm{j} } \\ \vec{ \rm{\mathbf{ f}} } + \vec{ \rm{\mathbf{ q}} } &=& \mathbf{ 12}\vec{ \rm{\mathbf{ i}} } \mathbf{ -3}\vec{ \rm{\mathbf{ j}} } \end{array}$$

33554432=x^5   x = ?

$$\small{\text{$ \rm{The~prime~ factorization~ of~} \mathbf{ 33554432 = 2^{25} } $}} \\ \small{\text{$ \rm{so~we~have~} \mathbf{ 2^{25} = x^5} $}} \\ \small{\text{$ \rm{or~} \mathbf{ 2^{5\cdot 5} = x^5} $}} \\ \small{\text{$ \rm{or~} \mathbf{ (2^5)^5 = (x)^5} \qquad | \qquad \rm {comparing~coefficient} $}} \\ \small{\text{$ \rm{so~} \mathbf{ (2^5) = x} $}} \\ \small{\text{$ \rm{or~} \mathbf{ 32 = x} $}} \\$$

Sally's brother Aaron is 15 years old. Carl is 8 years older than Aaron and the sum of their ages is 47 years. How old is sally? Show full algebraic working.

$$\small{\text{$ \begin{array}{rcl} \rm{Aaron~is~15~years~old}\\\\ \rm{Carl~=~8~years + Aaron}= 8 + 15 \\ \rm{Carl~is~23~years~old~}\\\\ \rm{Sally+Carl+Aaron}=47\\ \rm{Sally~}+23+15=47\\ \rm{Sally~}+38=47\\ \rm{Sally~}=47-38\\ \rm{Sally~}=9\\ \rm{Sally~is~9~years~old~} \end{array} $}}$$

heureka i still dont understand the brackets part

why :x^3+6x^2+12x+16  -> (x+4)(x^2+2x+4)

Hallo sabi92,

I.   $$\small{\text{$ x^3+6x^2+12x+16: $}}$$

   $$\small{\text{$ x = -4 \qquad \Rightarrow \qquad (-4)^3+6\cdot (-4)^2 + 12\cdot (-4) + 16 = -64 + 96 -48 + 16 = 0 $}}$$  

II.   algebraic division:

$$\small{\text{$ x^3+6x^2+12x+16~~:~~(x+4) =x^2+2x+4 $}}$$

Durch welche Primzahl kann ich 188 teilen,sodass ich keine Kommerzahlen erhalte (Primfaktorzerlegun)

I. Die Primfaktorzerlegung von 188:

$$188 = \underbrace{2^2}_{1. \rm{~Primzahl}=2 }~\cdot \underbrace{47}_{2. \rm{~Primzahl}=47 }$$

II. Die Primteiler von 188 sind die 2 und die 47:

1. 188 kann durch 2 (Primzahl) ohne Rest geteilt werden: $$\small{\text{$ \frac{188}{2}=94 $}}$$

2. 188 kann durch 47 (Primzahl) ohne Rest geteilt werden: $$\small{\text{$ \frac{188}{47}=4 $}}$$

$$\small{\text{$ \begin{array}{lrcl} (1):& \frac12 x+ \frac13 y+ \frac15 z &=& 71 \qquad | \qquad \cdot 30 \\\\ (2):& \frac13 x+ \frac15 y+ \frac12 z &=& 57 \qquad | \qquad \cdot 30 \\\\ (3):& \frac15 x+ \frac12 y+ \frac13 z &=& 58 \qquad | \qquad \cdot 30 \\\\ \hline &\\ (1):& 15x+ 10y+ 6z &=& 71\cdot 30 = 2130\\\\ (2):& 10x+ 6y+ 15z &=& 57\cdot 30 = 1710\\\\ (3):& 6x+ 15y+ 10z &=& 58\cdot 30 = 1740 \\\\ \hline &\\ \rm{Determinante~~} D= \begin{vmatrix} 15 & 10 & 6 \\ 10 & 6 & 15 \\ 6 & 15 & 10 \end{vmatrix} = -1891 &\\ x = \frac{ \begin{vmatrix} 2130 & 10 & 6 \\ 1710 & 6 & 15 \\ 1740 & 15 & 10 \end{vmatrix} }{D} = \dfrac{-170190}{-1891}=90 \\ &\\ y = \frac{ \begin{vmatrix} 15 & 2130 & 6 \\ 10 & 1710 & 15 \\ 6 & 1740 & 10 \end{vmatrix} }{D} = \dfrac{-113460}{-1891}=60 \\ &\\ z = \frac{ \begin{vmatrix} 15 & 10 & 2130 \\ 10 & 6 & 1710 \\ 6 & 15 & 1740 \end{vmatrix} }{D} = \dfrac{-56730}{-1891}=30 \\ \end{array} $}}$$

Y=4sec (1/2x)

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