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What are the sum (S) and product (P) of the roots of the equation 3x^2 - 7x + 12 = 0 ?

$$\small{\text{ $ \begin{array}{rcl} 3x^2 - 7x + 12 & = & 0 \quad | \quad :3 \\ \\ x^2 - \frac{7}{3}x + \frac{12}{3} & = & 0 \\ \\ x^2 \underbrace{- \frac{7}{3}}_{ = -S } x + \underbrace{ 4 }_{=P }& = & 0 \\\\ $ sum (S) of the roots: $x_1+x_2& = &+ \frac{7}{3} \\\\ $ product (P) of the roots: $ x_1*x_2& = & 4 \end{array} $ }}$$

 i dont know how to solve can someone help me

6x=2x^2

x=?

$$\small{\text{ $ \begin{array}{rcl} 6x & = & 2x^2 \\ 2x^2 - 6x &=& 0 \\ \underbrace{x}_{=0}(\underbrace{2x-6}_{=0}) &=& 0 \\ \end{array} $ }}$$

1. x = 0

2. 2x-6 = 0 | +6

    2x    = 6 | : 2

     x    = 6 /2

     x    = 3

 i dont know how to solve can someone help me

6x=2x^2

x=?

$$\small{\text{ $ \begin{array}{rcl} 6x & = & 2x^2 \\ 2x^2 - 6x &=& 0 \\ \underbrace{x}_{=0}(\underbrace{2x-6}_{=0}) &=& 0 \\ \end{array} $ }}$$

1. x = 0

2. 2x-6 = 0 | +6

    2x    = 6 | : 2

     x    = 6 /2

     x    = 3

fünf bauern brauchen insgesamt 12 tage um ein äpfelfeld komplett abzuernten wie viel tage brauchen es acht bauer ?

$$\small{\text{ \begin{array}{rcl} $5 $ Bauern $ *\ 12 $ Tage $ & = & 8$ Bauern $ * \ x $ Tage $= 1$ Apfelfeld \\ \\ 5 * 12 & = & 8 * x \quad | \quad : 8 \\\\ \dfrac{5*12}{8} &=& x \\\\ x &=& \dfrac{60}{8} \\\\ x &=& 7.5 \end{array} }}$$

8 Bauern brauchen nur 7.5 Tage.

 i dont know how to solve can someone help me

6x=x^2

x=?

$$\small{\text{ $ \begin{array}{rcl} 6x & = & x^2 \\ x^2 - 6x &=& 0 \\ \underbrace{x}_{=0}(\underbrace{x-6}_{=0}) &=& 0 \\ \end{array} $ }}$$

1. x = 0

2. x-6 = 0 | +6

    x    = 6

Any real Number multiplied by 0 is 0!

$$\infty * 0 \text{ is u n d e f i n e d and not 0 }$$ , because $$\infty \text{ is not a real number }$$

how to conert rectangle form 2+3i to polar form :

$$z = x + y i \quad | \quad z = 2 + 3i$$

polar: $$x = r*\cos{(\phi)} \quad and \quad y = r*\sin{(\phi)}$$

set into z = x + yi:   $$z = r*\cos{(\phi)}+ r*\sin{(\phi)}i = r \left[ \cos{(\phi)}+ \sin{(\phi)}i \right] = r \left[ \cos{(\phi)}+ i\sin{(\phi)} \right]$$

r:  $$r=\sqrt{x^2+y^2} \quad | \quad r = \sqrt{2^2+3^2} = \sqrt{4+9} = \sqrt{13}$$  

$$\phi$$  : $$\phi = \tan^{-1}{ ( \frac{y}{x} )} \quad | \quad \phi = \tan^{-1}(\frac{3}{2}) = 56.3099324740\ensurement{^{\circ}}$$

z(polar): $$z=r \left[ \cos{(\phi)}+ i\sin{(\phi)} \right] \quad | \quad z = \sqrt{13} \left[ \cos{(56.3099324740\ensurement{^{\circ}})}+ i*\sin{(56.3099324740\ensurement{^{\circ}})} \right]$$

$$\small{\text{ $ \begin{array}{rcl} f(x)=x+2 \\ f^{-1}(x) &=& ? \\ y & =& x+2 \quad | \quad x\leftrightarrow y \\ x & = & y+2 \\ y & = & x-2 \\ f^{-1}(x) & = & x-2 \\ \hline \\ g(x)=\dfrac{x}{3} \\ \\ g^{-1}(x) &=& ? \\ y & =& \dfrac{x}{3} \quad | \quad x\leftrightarrow y \\ \\ x & = & \dfrac{y}{3} \\ \\ y & = & 3x \\ g^{-1}(x) & = & 3x \\ \hline \end{array} $ }}$$

$$\small{\text{ $ \begin{array}{rcccl} f(19) & = & 19 + 2 & = & 21 \\ g(21) & = & \frac{21}{3} & = & 7 \\ f^{-1}(7) & = & 7-2 & = & 5 \\ f^{-1}(5) & = & 5-2 & = & 3 \\ g^{-1}(3) & = & 3*3 & = & 9 \\ f(9) & = & 9 + 2 & = & 11 \\ \end{array} $ }}$$

$$f(g^{-1}(f^{-1}(f^{-1}(g(f(19)))))) = 11$$

a+b=2 b-a=4

$$\begin{array}{ccc|c|c|} & & & I.& II.\\ b+a & = & 2 \ & &\\ & & & + & -\\ b-a & = & 4 \ & & \end{array}$$

I.

  $$\small{\text{ $ \begin{array}{rcl} 2b + a + (-a) & = & 2 +4 \\ 2b +0 & = &6 \\ 2b & = &6 \quad | \quad : 2\\ b & = & \frac{6}{2} \\ b &=& 3 \end{array} $ }}$$

II.

$$\small{\text{ $ \begin{array}{rcl} b - b + a - (-a) & = & 2 - 4 \\ 0+2a & = &-2 \\ 2a & = &-2 \quad | \quad : 2\\ a & = & \frac{-2}{2}\\ a &=& -1 \end{array} $ }}$$

Hi Melody,

$$\small{\text{ $ \begin{array}{|l|c|r|} \hline 7^0...7^{21}&& 1...22 \\ \hline 7^0 & mod\ 23 & 1 \\ \hline 7^1 & mod\ 23 & 7 \\ \hline 7^2 & mod\ 23 & 3 \\ \hline 7^3 & mod\ 23 & 21 \\ \hline 7^4 & mod\ 23 & 9 \\ \hline 7^5 & mod\ 23 & 17 \\ \hline 7^6 & mod\ 23 & 4 \\ \hline 7^7 & mod\ 23 & 5 \\ \hline 7^8& mod\ 23 & 12 \\ \hline 7^9& mod\ 23 & 15 \\ \hline 7^{10}& mod\ 23 & 13 \\ \hline 7^{11}& mod\ 23 & 22 \\ \hline 7^{12}& mod\ 23 & 16 \\ \hline 7^{13}& mod\ 23 & 20 \\ \hline 7^{14}& mod\ 23 & 2 \\ \hline 7^{15}& mod\ 23 & 14 \\ \hline 7^{16}& mod\ 23 & 6 \\ \hline 7^{17}& mod\ 23 & 19 \\ \hline 7^{18}& mod\ 23 & 18 \\ \hline 7^{19}& mod\ 23 & 11 \\ \hline 7^{\textcolor[rgb]{1,0,0}{20}}& mod\ 23 & \textcolor[rgb]{1,0,0}{8} \\ \hline 7^{21}& mod\ 23 & 10 \\ \hline \end{array} $ }}$$