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an AI you might have a use for.
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Chris
Posted 12/19/2024 09:36 (#11016672)
Subject: an AI you might have a use for.



East central Iowa

https://deepai.org/chat

Just don't ask political questions or social ones, it's rather biased.  But at math and some other things it's good.

For example I asked it what is  2^X + 8^X = 130

Here's it's reply:

To solve the equation (2^x + 8^x = 130), we start by rewriting (8^x) in terms of base 2:

[ 8^x = (2^3)^x = 2^{3x} ]

Now we can rewrite the equation:

[ 2^x + 2^{3x} = 130 ]

Next, let's set (y = 2^x). The equation becomes:

[ y + y^3 = 130 ]

Rearranging gives us:

[ y^3 + y - 130 = 0 ]

Now, we will find the roots of the cubic equation. One method is to try out possible rational roots. Testing (y = 5):

[ 5^3 + 5 - 130 = 125 + 5 - 130 = 0 ]

Thus, (y = 5) is a root. Next, we can factor the cubic polynomial. We can set up synthetic division with (y - 5):

      
5 | 1  0  1  -130  |    5  25  130------------------  | 1  5  26   0    

This tells us that:

[ y^3 + y - 130 = (y - 5)(y^2 + 5y + 26) ]

Next, we need to find the roots of the quadratic (y^2 + 5y + 26). The discriminant (\Delta) is computed as follows:

[ \Delta = 5^2 - 4 \cdot 1 \cdot 26 = 25 - 104 = -79 ]

Since the discriminant is negative, the quadratic (y^2 + 5y + 26) does not have real roots.

Thus, the only real solution comes from (y = 5):

[ 2^x = 5 ]

Taking the logarithm of both sides:

[ x = \log_2(5) ]

Thus, the solution to the equation (2^x + 8^x = 130) is:

[ x = \log_2(5) ]

 

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