Compute λ from α^2 + β^2: If α and β are roots of x^2 − 3λx + λ^2 = 0 and α^2 + β^2 = 7/4, find λ.

Introduction / Context:Given a quadratic in parameter λ, we use Vieta’s relations to express α + β and αβ in terms of λ. Then α^2 + β^2 is computed via the identity (α + β)^2 − 2αβ. Setting this equal to the provided value yields λ directly.

Given Data / Assumptions:

• x^2 − 3λx + λ^2 = 0 ⇒ α + β = 3λ, αβ = λ^2
• α^2 + β^2 = 7/4

Concept / Approach:Use α^2 + β^2 = (α + β)^2 − 2αβ. Substitute the expressions in λ and solve the resulting equation for λ. Note that squaring creates ± solutions for λ, which are acceptable when consistent.

Step-by-Step Solution:

Verification / Alternative check:Plug λ = ±1/2 back: α + β = ±3/2, αβ = 1/4. Then α^2 + β^2 = (±3/2)^2 − 2*(1/4) = 9/4 − 1/2 = 9/4 − 2/4 = 7/4, as required.

Why Other Options Are Wrong:

• ±√7/2, ±√3/2, ±1/3: Do not satisfy 7λ^2 = 7/4.
• None of these: Rejected since λ = ±1/2 works.

Common Pitfalls:Forgetting the factor 2 in 2αβ or mishandling signs when squaring. Always compute α^2 + β^2 via the identity to avoid expansion errors.

Final Answer:± 1/2