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2 (two)

The equation 2^x = x² can be solved by thinking carefully.

 

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Solution 1

For x = 2 you can see immediately that it is correct, because

2² = 2²     ⇒     4 = 4     ⇒     0 = 0

 

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Solution 2

Also x = 4 is a solution that you can guess quite easily, because

2⁴ = 4²     ⇒     16 = 16     ⇒     0 = 0

 

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Solution 3

If you draw a graph you see that there are three intersection points.

With a bit of dexterity, you can find the value x = −0.7667… on a calculator, because

2^−0,7667 = (−0,7667)²     ⇒     0,587... = 0,587...     ⇒     0 = 0

where this answer is accurate to three decimal places.

 

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Further explanation

Let's do something different, and take x→∞. Then you can write

          

That seems like a good answer, but something's not right here, because

∞ − ∞ =  ?

Infinity is not a number with a specific value. So there remain three solutions.

 

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