Shift roots by +2 (build new quadratic): If α and β are roots of x^2 − 11x + 24 = 0, find the quadratic equation whose roots are (α + 2) and (β + 2).

Introduction / Context:
Shifting roots by a constant k transforms the equation via the substitution y = x − k. If α, β are roots of f(y) = 0, then (α + k), (β + k) are roots of f(x − k) = 0. We use this to construct the new quadratic directly without solving for α and β numerically.

Given Data / Assumptions:

• Original: y^2 − 11y + 24 = 0 with roots α, β
• We need equation with roots α + 2, β + 2

Concept / Approach:
Set y = x − 2 so that when x = α + 2, we have y = α (and similarly for β). Substitute y = x − 2 into the original polynomial and expand to get the desired equation in x.

Step-by-Step Solution:

Verification / Alternative check:
Sum of new roots = (α + β) + 4 = 11 + 4 = 15 ⇒ coefficient of x is −15 (matches). Product = (α + 2)(β + 2) = αβ + 2(α + β) + 4 = 24 + 22 + 4 = 50 (constant term matches).

Why Other Options Are Wrong:

• Each alternative has incorrect sign or constant; only x^2 − 15x + 50 = 0 matches both the shifted sum and product.

Common Pitfalls:
Forgetting that shifting roots adjusts both sum and product: S′ = S + 2*2 and P′ = P + 2S + 4 for a shift of +2.

Final Answer:
x^2 − 15x + 50 = 0