User: Find me for which values of λ the equation λ*x^2 - 2*x +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