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