Roots differ by 1 (use Vieta and difference-of-roots identity): If α and β are roots of x^2 + kx + 12 = 0 with α − β = 1, find k.

Introduction / Context:For a quadratic x^2 + kx + 12 = 0, the sum and product of roots are S = −k and P = 12. The difference of roots satisfies (α − β)^2 = S^2 − 4P. Given α − β = 1, we can use this identity to determine S, and hence k, without explicitly solving for α and β.

Given Data / Assumptions:

• S = α + β = −k
• P = αβ = 12
• (α − β) = 1 ⇒ (α − β)^2 = 1

Concept / Approach:Use the identity (α − β)^2 = (α + β)^2 − 4αβ = S^2 − 4P. Set S^2 − 4P = 1 and solve for S. Then k = −S, which may yield two values corresponding to ±S.

Step-by-Step Solution:

Verification / Alternative check:Either k = 7 or k = −7 yields an equation whose roots differ by exactly 1. A quick plug with discriminant confirms that both choices produce real and distinct roots satisfying the difference condition.

Why Other Options Are Wrong:

• ±5, ±1, 0: These do not satisfy S^2 = 49 with P = 12, hence (α − β)^2 ≠ 1.

Common Pitfalls:Confusing α − β with |α − β|; here the square avoids sign issues. Also, do not attempt to solve the quadratic directly—Vieta’s identity is quicker and safer.

Final Answer:± 7