Roots where one is the square of the other: For x^2 − b x + c = 0, suppose one root is the square of the other. Identify the correct relation between b and c.

Introduction / Context:
Let the roots be r and r^2 (order irrelevant). Then the sum and product must match Vieta’s relations for x^2 − bx + c = 0: r + r^2 = b and r^3 = c. Eliminating r between these two relations yields a polynomial identity connecting b and c only.

Given Data / Assumptions:

• Sum: r + r^2 = b
• Product: r^3 = c
• r is real (implied by a standard quadratic context), but the identity holds algebraically.

Concept / Approach:
Express r^2 as b − r and compute r^3 = r * r^2 = r(b − r) = br − r^2. Replace r^2 again by (b − r) to get r^3 in terms of r and b alone. Then substitute c for r^3 and eliminate r via the earlier linear relation to reach a pure b–c identity. A quick numeric check confirms the final formula.

Step-by-Step Solution:

Verification / Alternative check:
Test r = 2 ⇒ b = 2 + 4 = 6, c = 8. Then b^3 = 216 and 3bc + c^2 + c = 3*6*8 + 64 + 8 = 216, confirming the relation.

Why Other Options Are Wrong:

• They do not hold for sample values (e.g., r = 2) and conflict with the derived identity from Vieta’s formulas.

Common Pitfalls:
Trying to solve for r explicitly and then back-substitute with approximate decimals. Work symbolically to maintain exactness.

Final Answer:
b^3 = 3bc + c^2 + c