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KCC's Quizzes AQQ253 about a funny equation

And here is a new challenge! It's easier than the previous one but still...

Please spread those weekly quizzes among your colleagues!

Also, please be informed awards and goodies are being sent to the frequent winners. Please apologize for the long delay since we do that only 2 times per year!

 Certificates of the wins are also sent in paper form; the ones desiring a digital copy, please let us know!

Some statistics about the frequent winners by country origins: (we have also statistics by individuals, but they can be provided on demand and only with own data).

Regards

Kuo-Chang

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  • same comment as above...

  • So we can get value of t as (-5+i√79) /2 and  (-5-i√79) /2 so value of x will be log( -5+i√79) /2*log2 and  (-5-i√79) /2*log2.

    Not sure if we can calculate further. 

    If we can please help me know more about that. 

  • Use the Euler transformation:   a + b i   =  r*exp(i*t)    with r^2  = a^ + b^2   and t is the angle with a base "a" and a rise "b". 
                                                     
    So  ln(a+bi)  =  ln( r*exp(it)) = ln(r) + ln(exp( i*t ))  =  ln(r)  +  i*t  

    (Euler transform is like to transform a point to polar coordinate  from its cartesian coordinates (real and imaginary axis))

  • If you calculate x from the complex solutions you express the numbers as |mag| times e^(iTheta).  Then you use the natural log (ln) to solve for x.  This results in an infinite number of answers separated by 2Pi.

  • Tahnks  for your reply!

    Yes, 2.321925 is well a solution to the equation, congratulations! BUT it is not the only one; there are 2 other ones! May be you can find them?

  • Yes! @Sina22! You get it. The expressions fo the 2 complex values can be simplified as -2.5 +/- i 4.4444

    Congratulation!

  • And it is time to publish the official answers for this quiz about the funny equation. In fact most of you have found the real solution which is Ln(5)/Ln(2) = 2.322. BUT the question is about ALL the possible solutions! And there are solutions that are in the complex world.

    Here below is the detailed development:

    Let’s rewrite the equation as:

    (23)x + 2x = 130

                      Or:    (2x)3 + 2x = 130 and lets pose 2x = y

                      Therefore: y3 + y = 130

                      One can observe 130 = 26 * 5

                      BY adding and substracting 26y, the equation can be rewritten as:

                               y3 + y + 25y – 25y = 26*5

                      Hence: y3 + 26y – 25y - 26*5 = 0

                      By making specific grouping: (y3 – 25y) + (26y - 26*5) = 0

                      Hence: y(y²-25) +26(y-5)=0  or y(y-5)(y+5) + 26(y-5)=0

                      (y-5) (y(y+5)+26=0 or (y-5) (y²+5y+26)=0

                      The above equation has 3 solutions for y; 1 real and 2 complex

                      The real solution is y=5 giving x = Ln(5)/Ln(2) = 2.322

                      The complex solutions are given by y²+5y+26 = 0

                                        y = [-5 +/-SQRT(25-4*26)]/2 = [-2.5 +/-0.5*SQRT(-79)]

                                       y = [-2.5 +/-0.5*i*8.888] = (-2.5 +/-i*4.444)

    Congratulation to all having found at least the real solution and big applause to the ones having found the 3 solutions  from Cavite/Philippines, @retiredEE from USA and  from India !

    Now, be all prepared for the next coming quiz which will be more dedicated to our electronic engineers...

    P.S. : there are goodies and certificate distributed to all the frequent winners. If you think you should be part of them and you don't get any reward, please let me know.

  • Fully correct  ! You found the real solution, but there are 2 complex solutions as well...