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

  • MHB
  • Thread starter late6002
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In summary, the equation 2^x+ 2/2^x= 3 can be reduced to the quadratic equation y^2- 3y+ 2= 0 by substituting y= 2^x. This equation can then be factored into (y- 2)(y- 1)= 0, giving solutions of x= 0 and x= 1. By substituting these values back into the original equation, we can verify that they are indeed solutions.
  • #1
late6002
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2^x + 2/2^x =3
 
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  • #2
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
 
  • #3
Hi late6002, welcome to MHB!

For your information, the cited equation can be reduced to quadratic equation...
 
  • #4
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.
 

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

What is the equation 2^x + 2/2^x = 3?

The equation 2^x + 2/2^x = 3 is a mathematical equation that involves an exponent and a fraction. It is commonly used in algebra and can be solved for the value of x.

What is the purpose of solving 2^x + 2/2^x = 3?

The purpose of solving 2^x + 2/2^x = 3 is to find the value of x that makes the equation true. This can help in understanding the relationship between exponential and fractional expressions.

What are the steps to solve 2^x + 2/2^x = 3?

The steps to solve 2^x + 2/2^x = 3 are as follows:

  • Combine the terms with the same base of 2^x by using the exponent rule of addition.
  • Multiply both sides by 2^x to eliminate the fraction.
  • Simplify the equation to get 2^(2x) - 2 = 3(2^x).
  • Substitute y = 2^x to get a quadratic equation of y^2 - 3y - 2 = 0.
  • Solve for y using the quadratic formula.
  • Substitute back in for x to get the solutions for the original equation.

What are the possible solutions for 2^x + 2/2^x = 3?

The possible solutions for 2^x + 2/2^x = 3 are x = 1 and x = -1. These values can be found by solving the quadratic equation y^2 - 3y - 2 = 0 and substituting back in for x.

How can solving 2^x + 2/2^x = 3 be applied in real life?

Solving 2^x + 2/2^x = 3 can be applied in real life situations involving exponential and fractional relationships. For example, it can be used in finance to calculate compound interest, in science to model population growth, or in engineering to determine the growth rate of a certain material.

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