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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