# Example that help for Improvement of system in solving Math problems

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

(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

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