Example that help for Improvement of system in solving Math problems

ChatGPT
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User: Find me for which values of λ the equation λx^2 - 2x +15*λ = 0 has real roots. Verify using specific roots from the set you found.

ChatGPT: The equation λx^2 - 2x +15*λ = 0 has real roots if and only if the discriminant is greater than or equal to zero, that is:

Δ = b^2 - 4ac = (-2)^2 - 4λ(15λ) = 4 - 60λ^2

Thus, we must have:

4 - 60λ^2 >= 0

From the above inequality, we find that:

-√(15/4) <= λ <= √(15/4)

Therefore, the equation has real roots for every value of λ in the interval [-√(15/4), √(15/4)].

For example, if we set λ = 1, then the equation becomes:

x^2 - 2x + 15 = 0

The distinguishing feature is:

Δ = (-2)^2 - 4(1)(15) = -56

Because the discriminant is negative, the equation has no real roots.

Conversely, if we set λ = 1/2, then the equation becomes:

(1/2)x^2 - 2x + (15/2) = 0

The distinguishing feature is:

Δ = (-2)^2 - 4(1/2)(15/2) = 16 - 30 = -14

Again, the equation has no real roots.

Finally, if we set λ = √(15)/4, then the equation becomes:

(√15/4)x^2 - 2x + (15√15)/4 = 0

The distinguishing feature is:

Δ = (-2)^2 - 4(√15/4)(15√15/4) = 4 - 225/4 = -209/4

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Example that help for Improvement of system in solving Math problems

Welcome @starsee

Sorry to disappoint you, but ChatGPT does not “solve” anything. ChatGPT is a language model and it simply generates next sequence tokens based in the current token(s). It generates this text from a large language model will billions (trillions) of data from the public internet (book, etc).

So ChatGPT does not “solve” any math, it just generates text, like a fancy auto-completion engines, predicting text based on probability.

If I run your same prompt with ChatGPT, I get a totally different completions because ChatGPT is just “babbling” out text based on probability as a language model.

prompt: Find me for which values of λ the equation λx^2 - 2x +15*λ = 0 has real roots. Verify using specific roots from the set you found.

completion:

For the quadratic equation λx^2 - 2x + 15λ = 0 to have real roots, its discriminant (b^2 - 4ac) must be greater than or equal to zero. In this case, we have:

b^2 - 4ac = (-2)^2 - 4(λ)(15λ) = 4 - 60λ^2

So, for the equation to have real roots, we must have:

4 - 60λ^2 ≥ 0

Simplifying, we get:

60λ^2 ≤ 4

15λ^2 ≤ 1

-1/15 ≤ λ ≤ 1/15

Thus, the equation λx^2 - 2x + 15λ = 0 has real roots for λ values between -1/15 and 1/15, inclusive.

To verify, we can choose a specific value of λ within this range and find the corresponding roots. Let's take λ = 1/30, which is a valid value within the range we found.

Plugging λ = 1/30 into the original equation, we get:

(1/30)x^2 - 2x + (15/2) = 0

Multiplying both sides by 30 to eliminate the fraction, we get:

x^2 - 60x + 225 = 0

This quadratic equation factors as:

(x - 15)(x - 15) = 0

So the only root is x = 15, which is a real number.

Therefore, we have confirmed that for λ = 1/30, the equation λx^2 - 2x + 15λ = 0 has a real root.

We all understand people enjoy playing and experimenting with ChatGPT and math, physics or other technical problems, but please keep in my that ChatGPT is just generating text like a probabilistic auto-completion engine based on a massive scale large language model and ChatGPT is not “solving” anything.

HTH