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heureka

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 #1
avatar+26396 
+2

Hi - what is the best way to solve simultaneous equations where one is quadratic and the other linear?

For example:

y=x+3

y=x2+3x

 

1. Formula line ( linear equation)

yline=mxline+b

 

2. Formula parabola (quadatic equation)

yparabola=Ax2parabola+Bxparabola+C

 

3. set equal  yline=yparabola=yintersection :

mxintersection+b=Ax2intersection+Bxintersection+CAx2intersection+xintersection(Bm)+Cb=0xintersection1,2=mB±(mB)24A(Cb)2Ayintersection1,2=mxintersection1,2+b

 

4. Example:

y=x+3m=1b=3y=x2+3xA=1B=3C=0xintersection1,2=13±(13)241(03)21xintersection1,2=2±4+122xintersection1,2=2±42xintersection1=2+42=1xintersection2=242=3yintersection1=1xintersection1+3=11+3=4yintersection2=1xintersection2+3=1(3)+3=0

 

laugh

11.04.2017
 #3
avatar+26396 
+3

Integrate the following with steps please:

integrate [sin^2 (x) cos^2 (x)] dx from x=0 to pi.

Thank you for any help.

 

Formula:

cos2(x)=12(1+cos(2x))sin2(x)=12(1cos(2x))

 

 

π0sin2(x)cos2(x) dx=π012(1cos(2x))12(1+cos(2x)) dx=14π0(1cos(2x))(1+cos(2x)) dx=14π0(1cos2(2x)) dx|cos2(2x)=12(1+cos(4x))=14π0(112(1+cos(4x)) dx=14π0(112cos(4x)) dx=14π0(12cos(4x)) dx=14[x2sin(4x)4]π0=18[xsin(4x)2]π0=18[πsin(4π)(0sin(40)2)]=18[π0(00)]=18[π]=π8

*edited

 

laugh

11.04.2017
 #20
avatar+26396 
+3

How many 3s..........

 

How many digits of 3 are there between 1 and 10^12, and what is the percentage of these 3's out of the total of 10^12 numbers, and why? Please explain.

Thank you for any help.

 

I agree with Alan the total count of 3s is in general 

between 1 and 10n n10n1

between 1 and 1012  = 121011

 

My empirical solution:

n12th11th10th9th8th7th6th5th4ththirdsecondfirstSum of 3's3's in Postion11011×311100210×310×320110003100×3100×3100×330011000041000×31000×31000×31000×340001100000000000012100000000000×3100000000000×3100000000000×3100000000000×3100000000000×3100000000000×3100000000000×3100000000000×3100000000000×3100000000000×3100000000000×3100000000000×31200000000000

 

laugh

11.04.2017
 #3
avatar+26396 
+2

I want to caculate 11power 850 mod 1643 

 

11850(mod1643)114212+2(mod1643)(114)212121(mod1643)|114(mod1643)146(mod1643)(146)212121(mod1643)(146)2106121(mod1643)[(146)2]106121(mod1643)|(146)2(mod1643)43(mod1643)(43)106121(mod1643)(43)426+2121(mod1643)[(43)4]26(43)2121(mod1643)|(43)4(mod1643)282(mod1643)(282)26(43)2121(mod1643)|(43)2(mod1643)206(mod1643)(282)26206121(mod1643)(282)213206121(mod1643)[(282)2]13206121(mod1643)|(282)2(mod1643)660(mod1643)66013206121(mod1643)66026+1206121(mod1643)(6602)6660206121(mod1643)|6602(mod1643)205(mod1643)2056660206121(mod1643)|660206121(mod1643)199(mod1643)2056(199)(mod1643)20523(199)(mod1643)(2052)3(199)(mod1643)|2052(mod1643)693(mod1643)(693)3(199)(mod1643)(693)2+1(199)(mod1643)(693)2(693)(199)(mod1643)|(693)(199)(mod1643)105(mod1643)(693)2(105)(mod1643)|(693)2(mod1643)493(mod1643)493(105)(mod1643)493(105)(mod1643)|493(105)(mod1643)832(mod1643)811(mod1643)811(mod1643)

 

laugh

10.04.2017