What is number of roots of equation 4^x+2^x=272?

Expert Answers
sciencesolve eNotes educator| Certified Educator

You should come up with the following substitution, such that:

`2^x = t`

Replacing the variable in equation, yields:

`t^2 + t = 272 => t^2 + t - 272 = 0`

Using quadratic formula, yields:

`t_(1,2) = (-1+-sqrt(1 + 1088))/2 => t_(1,2) = (-1+-33)/2`

`t_1 = (-1+33)/2 => t_1 = 16`

`t_2 = (-1-33)/2 => t_2 = -17`

You need to solve for x the equations `2^x = t_1` and `2^x = t_2` , such that:

`2^x = t_1 => 2^x = 16 => 2^x = 2^4 `

Equating the powers yields:

`x = 4`

`2^x = t_2 => 2^x = -17` invalid because `2^x > 0`

Hence, evaluating the number of solutions to the given equation, yields one solution, `x = 4.`

degeneratecircle eNotes educator| Certified Educator

Another method is to use the fact that `f(x)=4^x+2^x` is an increasing function. If you're not expected to know calculus, then you can probably assume this, maybe by appealing to the graph.

If you know calculus, then this isn't too difficult to prove using derivatives.

Once you know it's an increasing function, then there can be at most one point where `f(x)=272.` Again by using calculus (specifically the Intermediate Value Theorem), or by trying small numbers until one works, we can say that there is exactly one value of `x` for which `4^x+2^x=272.`

aruv | Student


rewrite equation as



Let `2^x=y` then above equation reduces to


which is quadratic equation.So it has two roots which  may be real /complex.




`or y=16`


`2^x=-17`  which is not possible for any value of x.




Thus it has one real root and other root not possible.