MHB What are the solutions to the equation 2^x + 2/2^x = 3?

  • Thread starter Thread starter late6002
  • Start date Start date
late6002
Messages
1
Reaction score
0
2^x + 2/2^x =3
 
Mathematics news on Phys.org
In future kindly inform what you have tried and where you are stuck so that we can provide steps to proceed

For this put $2^x = y$ and see what you get
 
Hi late6002, welcome to MHB!

For your information, the cited equation can be reduced to quadratic equation...
 
Since this has been here over a month now (and I just can't resist answering):

The equation is 2^x+ 2/2^x= 3. Following Kaliprasad's advice, let y= 2^x. Then the equation becomes, y+ 2/y= 3. Multiply both sides by y to get y^2+ 2= 3y. As anemone said, that is a quadratic equation, y^2- 3y+ 2= 0. And that is easy to factor: (y- 2)(y- 1)= 0. Either y= 2 or y= 1. Since y= 2^x, if y= 2, 2= 2^x so x= 1. If y= 1, 1= 2^x so x= 0.

Check: if x= 0, 2^x= 2^0= 1 so 2^x+ 2/2^x= 1+ 2/1= 1+ 2= 3. If x= 1, 2^x= 2^1= 2 so 2^x+ 2/2^x= 2+ 2/2= 2+ 1= 3.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top