Solve for x when it's in a polynomial and exponent

[dehgtw32]

How do you solve for x in a question that are like these (without using a graphing calculator)? I think it has to do with logarithmic properties but that's the best I can figure.

(a) (0.5)^x - 2 = -x^2 + 4

(b) 2^x - x^2

(c) (0.5) ^ x - x ^ -1, x is not 0

(d) 2 ^ x = - x ^ 2 + 3x + 2

(e) 5x ^ 2 = 3 ^ x

(f) 2 ^ x = x ^ 3 + x ^ 2 - 6x

(g) 2 ^ x + 0.5 = 4 - x ^ 2

 

[Deleted member 4993]

How do you solve for x in a question that are like these (without using a graphing calculator)? I think it has to do with logarithmic properties but that's the best I can figure.

(a) (0.5)^x - 2 = -x^2 + 4 ... are you sure the problem does not look like (0.5)^{x - 2} = -x^2 + 4 which should be written as (0.5)^(x - 2) = -x^2 + 4

(b) 2^x - x^2 ............ cannot solve for 'x' - there is no "=" sign

(c) (0.5) ^ x - x ^ -1, x is not 0 ............ cannot solve for 'x' - there is no "=" sign

(d) 2 ^ x = - x ^ 2 + 3x + 2

(e) 5x ^ 2 = 3 ^ x

(f) 2 ^ x = x ^ 3 + x ^ 2 - 6x

(g) 2 ^ x + 0.5 = 4 - x ^ 2

These problems - as far as I can see - can be solved only approximately, using numerical methods (e.g. Newton's method).

 

[dehgtw32]

Making progress

This isn't a regular quadratic equation so the standard rules for solving ax^2 + bx + c won't work. That's why I was so confused.

I also know the maximum solutions for x is the same the highest polynomial, x^3 could have 3 solutions but also less, etc.

I found f(x) and f'(x) to use the Newton's method. Now I'm just not sure how to determine the exact number of solutions if I don't know the shape of the graph or how to approximate x0.

 

[Deleted member 4993]

Knowing the function, whether monotone or not, can give you idea about the number of zeroes. You can also use your graphing calculator (or wolframalpha.com) to graph the function.

 

[dehgtw32]

Calculator

I currently don't have a graphing calculator and was just using a regular scientific calculator. I was hoping to avoid purchasing one. For homework it's easy to go online and plug it into a site that will graph it for me but that doesn't work so well for tests and exams. That's why I needed a solution method to solve these questions in the first place. Any suggestions are greatly appreciated.

 

[Dale10101]

Reflecting

I currently don't have a graphing calculator and was just using a regular scientific calculator. I was hoping to avoid purchasing one. For homework it's easy to go online and plug it into a site that will graph it for me but that doesn't work so well for tests and exams. That's why I needed a solution method to solve these questions in the first place. Any suggestions are greatly appreciated.

Speaking from reflection rather then authority:

http://www.regentsprep.org/Regents/math/algtrig/ATE8/exponentialEquations.htm is a link that demonstrates the basic types of exponential equations that can be solved analytically.

The difficulty with the problems that you present seems to be that there is no property that allows taking the log (inverse function) of, in general, a polynomial and express it an exponent of a base.

http://tutorial.math.lamar.edu/Classes/CalcI/NewtonsMethod.aspx is a link that explains Newton's method and demonstrates with a couple of examples if that is helpful. It does seem that you need to graph an equation if you want to get in the ball park, so to speak, before using Newton's method.

I did find it interesting to graph each side of some of your equations to get a better idea of when they will zero each other out.

What class are you taking anyway that they present such problems ... Calculus, applications of derivatives?