What are the three real solutions for the equation x^2=2^x?

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SUMMARY

The equation x^2=2^x has three real solutions, which can be identified by analyzing the intersection points of the curves y=x^2 and y=2^x. The known solutions are x=2 and x=4, while the third solution is negative and requires numerical methods for precise determination. Newton's Method and the Bisection Method are both effective techniques for finding this third root, with the latter being simpler for initial estimates.

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Homework Statement


Consider the equation x^2=2^x. Show that there are three real solutions by sketching appropriate curves. Solve the equation.



Homework Equations


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The Attempt at a Solution


Not really a calculus style question but because it is part of my calculus work I included it here. The first part of this question was simple: I sketched the graph of this equation, finding out that there are 3 points of intersection. The problem I'm having is actually finding what these intersection values are. I know just from looking at the graph and using guess and check that 2 and 4 are two of the values, but I have no idea how to find the third (negative) value. For some reason I was thinking of using Newton's Method, but I do not know how I could apply it to this question, nor if that is even the right method to use to find the value. Any nudge in the right direction for this question would be greatly appreciated, thanks.
 
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You are already going the right direction. You need to apply a numerical method to find the third root. The root is a zero of the function f(x)=x^2-2^x. Newton's method will work, but it's maybe a bit complicated. You could also use bisection to get a quick estimate.
 

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