Questions   
Sort: 
 #1
avatar+448 
+7
Oct 18, 2019
 #21
avatar+2440 
+3

TPM, I was wondering when you were going to show up again. Your appearances have a definite correlation to the mushrooms that suddenly appear on cow manure. ...And the psychotropic effects are similar too.   This time your appearance is related to math instead of advocacy for morons and brain-dead vandals.   ... It looks like you’ve continued to improve your math skills.  Your final equation is impressive –I’m still playing with it. (It could stand to have more annotation.)  It’s much better than your delusional, chaotic, bullshit infused presentation here.

 

LancelotLink sent me similar –well annotated, equations showing the relationships between Stirling Numbers of the First and Second kinds to Lah numbers (aka Stirling Numbers of the Third kind), and to Bell numbers. The annotations included footnotes showing applications in physics where in certain mediums, electromagnet resonance produces harmonic convergences that become soliton waves reflecting in some time-domains while passing through others.  A similar phenomenon also occurs for acoustical resonance in certain mediums. All of these are related to the geometry and hypergeometry of the medium. Even though most of the math was and still is beyond my skills, I thought it quite cool that that these relationships casually occur in nature and that someone could actually discover them.

 

Your equations are not anything I could casually produce, but I can understand them if I invest the time. I’m assuming they are your equations –the wagging head and jointless gestures are readily apparent, but it lacks the long monologues (aka Blarney), and... Perhaps the computer knows something, because it ... (shows the comment as if I didnt write it). ... so ...if not, could you give a source? 

 

GA

Oct 18, 2019
 #2
avatar
+1
Oct 18, 2019
 #2
avatar
0
Oct 18, 2019
 #1
avatar+26376 
+4

Simplify

\(\dfrac{6^x\cdot 12^{x+1}\cdot 20^{x+2}\cdot 30^{x+3}\cdot 42^{x+4}\cdot 56^{x+5}\cdot 72^{x+6}\cdot 90^{x+7} }{10!\cdot 8^x\cdot 15^{x+1}\cdot 24^{x+2}\cdot 35^{x+3}\cdot 48^{x+4}\cdot 63^{x+5}\cdot 80^{x+6}} \)

 

\(\begin{array}{|rcll|} \hline && \mathbf{ \dfrac{6^x\cdot 12^{x+1}\cdot 20^{x+2}\cdot 30^{x+3}\cdot 42^{x+4}\cdot 56^{x+5}\cdot 72^{x+6}\cdot 90^{x+7} }{10!\cdot 8^x\cdot 15^{x+1}\cdot 24^{x+2}\cdot 35^{x+3}\cdot 48^{x+4}\cdot 63^{x+5}\cdot 80^{x+6}} } \\\\ &=& \dfrac{(2\cdot3)^x\cdot (3\cdot4)^{x+1}\cdot (4\cdot5)^{x+2}\cdot (5\cdot6)^{x+3}\cdot (6\cdot7)^{x+4}\cdot (7\cdot8)^{x+5}\cdot (8\cdot9)^{x+6}\cdot (9\cdot10)^{x+7} } {10!\cdot (2\cdot4)^x\cdot (3\cdot5)^{x+1}\cdot (4\cdot6)^{x+2}\cdot (5\cdot7)^{x+3}\cdot (6\cdot8)^{x+4}\cdot (7\cdot9)^{x+5}\cdot (8\cdot10)^{x+6}} \\\\ &=& \dfrac{2^x3^x3^{x+1}4^{x+1}4^{x+2}5^{x+2}5^{x+3}6^{x+3}6^{x+4}7^{x+4}7^{x+5}8^{x+5}8^{x+6}9^{x+6}9^{x+7}10^{x+7} } {10!2^x4^x3^{x+1}5^{x+1}4^{x+2}6^{x+2}5^{x+3}7^{x+3}6^{x+4}8^{x+4}7^{x+5}9^{x+5}8^{x+6}10^{x+6}} \\\\ &=& \dfrac{3^x4^{x+1}5^{x+2}6^{x+3}7^{x+4}8^{x+5}9^{x+6}9^{x+7}10^{x+7} } {10!4^x5^{x+1}6^{x+2}7^{x+3}8^{x+4}9^{x+5}10^{x+6}} \\\\ &=& 3^x9^{x+7} \dfrac{4^{x+1}5^{x+2}6^{x+3}7^{x+4}8^{x+5}9^{x+6}10^{x+7} }{10!4^x5^{x+1}6^{x+2}7^{x+3}8^{x+4}9^{x+5}10^{x+6}} \\\\ &=& 3^x9^{x+7} \dfrac{4^{x+1-x}5^{x+2-x-1}6^{x+3-x-2}7^{x+4-x-3}8^{x+5-x-4}9^{x+6-x-5}10^{x+7-x-6} }{10!} \\\\ &=& 3^x9^{x+7} \dfrac{4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10 }{10!} \\\\ &=& 3^x9^{x+7} \dfrac{4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10 }{3!4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10} \\\\ &=& \dfrac{3^x9^{x+7}} {3!} \\\\ &=& \dfrac{3^x9^{x+7}} {2\cdot 3} \\\\ &=& \dfrac{1}{2}\cdot 3^{x-1}9^{x+7} \\\\ &=& \dfrac{1}{2}\cdot 3^{x-1}3^{2(x+7)} \\\\ &=& \dfrac{1}{2}\cdot 3^{x-1+2(x+7)} \\\\ &=& \mathbf{\dfrac{1}{2}\cdot 3^{3x+13}} \\ \hline \end{array}\)

 

laugh

Oct 18, 2019
 #1
avatar+26376 
+4

Find the values of k for which \(2x^2 + {\color{red}k}xy + 3y^2 - 5y - 2\) factors into two linear factor.

Source: https://www.askiitians.com/iit-jee-algebra/quadratic-equations-and-expressions/resolution-of-a-quadratic-function-into-linear-factors.aspx

 

In general:

\(\begin{array}{|lrcll|} \hline \text{Let } f(x, y) = ax^2 + 2hxy + by^2 + 2gx + 2fy + c \\ \text{writing this in descending powers of } y \text{ and equating to zero, we have } \\ by^2 + 2(hx + f)y + (ax^2 + 2gx + c) = 0 \\\\ \text{this is a quadratic equation in } y. \\ \text{Solving this for } y \text{, we get } y = \dfrac{ - (hx+f) \pm \sqrt{ (hx+f)^2 -b(ax^2+2gx+c)} } {b} \\ \hline \end{array} \)

 

\(\begin{array}{|lrcll|} \hline \text{The quantity under radical must be perfect square, }\\ \text{which is quadratic in }x\text{ and for the desired result, its discriminant must be zero: } \\\\ (hx+f)^2 -b(ax^2+2gx+c) \\ = h^2x^2+2hfx+f^2-abx^2-2bgx-bc \\ = (h^2-ab)x^2+2(hf-bg)x+(f^2-bc) \\\\ x = \dfrac{-2(hf-bg)\pm \sqrt{(-2(hf-bg))^2-4(h^2-ab)(f^2-bc)} } {2(h^2-ab)} \\ x = \dfrac{-2(hf-bg)\pm \sqrt{(4(hf-bg)^2-4(h^2-ab)(f^2-bc)} } {2(h^2-ab)} \\ x = \dfrac{-2(hf-bg)\pm 2\sqrt{(hf-bg)^2- (h^2-ab)(f^2-bc)} } {2(h^2-ab)} \\\\ (hf-bg)^2- (h^2-ab)(f^2-bc) = 0 \\ h^2f^2-2hfbg+b^2g^2-h^2f^2+h^2bc+abf^2-ab^2c = 0 \\ -2hfbg+b^2g^2 +h^2bc+abf^2-ab^2c = 0 \quad | \quad : b \\ -2hfg+bg^2 +h^2c+af^2-abc = 0 \quad | \quad \cdot (-1) \\ 2hfg-bg^2 -h^2c-af^2+abc = 0 \\ \mathbf{abc+2hfg-af^2-bg^2-h^2c = 0} \quad | \text{ The condition of two linear factors } \\ \hline \end{array} \)

 

\(\begin{array}{|rcll|} \hline f(x, y) = ax^2 + 2hxy + by^2 + 2gx + 2fy + c \\ \hline 2x^2 + {\color{red}k}xy + 3y^2 - 5y - 2 \\ \text{Here, } a=2,\ 2h={\color{red}k},\ b=3, g=0,\ 2f = -5,\ c= -2 \\\\ \mathbf{abc+2hfg-af^2-bg^2-h^2c} &=& \mathbf{0} \\ 2\cdot 3\cdot(-2)+k\cdot\left( \dfrac{-5}{2} \right)\cdot 0 -2\cdot \left( \dfrac{-5}{2} \right)^2 - 3\cdot 0^2 -\left( \dfrac{k}{2} \right)^2\cdot(-2) &=& 0 \\ -12 -\dfrac{25}{2} +\dfrac{k^2}{2} &=& 0 \quad | \quad \cdot 2 \\ -24 - 25 + k^2 &=& 0 \\ -49 + k^2 &=& 0 \\ k^2 &=& 49 \\ \mathbf{k} &=& \mathbf{\pm 7} \\ \hline \end{array}\)

 

Result:

\(\begin{array}{|lrcll|} \hline k = 7: \\ & (2x+y-2)(x+3y+1)&=& 2x^2+6xy+2x+xy+3y^2+y-2x-6y-2 \\ &&=& 2x^2+{\color{red}7}xy+3y^2-5y-2 \\\\ k = -7: \\ & (2x-y+2)(x-3y-1)&=& 2x^2-6xy-2x-xy+3y^2+y+2x-6y-2 \\ &&=& 2x^2{\color{red}-7}xy+3y^2-5y-2 \\ \hline \end{array} \)

 

laugh

Oct 18, 2019

2 Online Users

avatar
avatar