If one root of the quadratic equation A x^2 + B x + C = 0 is two and a half times the other root, which of the following relationships between A, B and C must be true?

Introduction / Context:
This algebra question explores the relationship between the coefficients of a quadratic equation and its roots. We are told that one root is two and a half times the other, and we must find a constraint on the coefficients A, B and C of the equation A x^2 + B x + C = 0 that reflects this ratio. The problem uses Vieta formulas, which connect the sum and product of the roots to the coefficients of the quadratic.

Given Data / Assumptions:

• Quadratic equation: A x^2 + B x + C = 0.
• Let the roots be r and (5/2) r, since one root is 2.5 times the other.
• We must find a true relationship among A, B and C from the options given.
• A, B and C are real coefficients, and A ≠ 0.

Concept / Approach:
For a quadratic A x^2 + B x + C = 0 with roots α and β, Vieta formulas state: α + β = -B / A, αβ = C / A. In this problem, we set α = r and β = (5/2) r. We can express the sum and product of the roots in terms of r and then relate them to -B / A and C / A. Eliminating r will give a condition involving only A, B and C. That condition can then be compared with the answer choices.

Step-by-Step Solution:
Step 1: Let the smaller root be r, then the larger root is (5/2) r. Step 2: Use Vieta formulas to express the sum and product of the roots. Sum of roots: r + (5/2) r = (7/2) r = -B / A. Product of roots: r * (5/2) r = (5/2) r^2 = C / A. Step 3: Express r^2 from the product relation: (5/2) r^2 = C / A ⇒ r^2 = (2C) / (5A). Step 4: Square the sum relation: (7/2)^2 r^2 = B^2 / A^2. That is (49/4) r^2 = B^2 / A^2. Step 5: Substitute r^2 from Step 3 into this equation. (49/4) * (2C / (5A)) = B^2 / A^2. Step 6: Simplify the left side: (49 * 2C) / (4 * 5A) = (98C) / (20A) = (49C) / (10A). So B^2 / A^2 = (49C) / (10A). Step 7: Multiply both sides by A^2 to eliminate denominators: B^2 = (49C / 10A) * A^2 = (49 C A) / 10. Step 8: Multiply both sides by 10: 10 B^2 = 49 C A.

Verification / Alternative check:
As a check, suppose A, B and C satisfy 10B^2 = 49 C A. We can choose convenient values for A and C that satisfy this relation and then solve the quadratic to confirm that the roots are indeed in the ratio 5 : 2. For example, setting A and C so that the relation holds produces a specific equation whose roots can be computed numerically. The ratio coming out as 2.5 provides strong confirmation of the derived condition.

Why Other Options Are Wrong:
Options a, b and c involve 7B^2 equal to various multiples of C A, which would correspond to different ratios between the roots. Option e, 36B^2 = 49 C A, also represents a different relationship and does not follow from the combination of Vieta formulas with the ratio (5/2). Only option d, 10B^2 = 49 C A, emerges from a consistent elimination of r and matches the exact algebraic steps shown above.

Common Pitfalls:
A common mistake is to misinterpret the phrase two and a half times as 2 times plus 1/2, leading to incorrect expressions for the roots. Another error is to forget to square the sum correctly or to misapply Vieta formulas, for example confusing the sign in -B / A. Some students attempt to guess the relationship by intuition instead of solving systematically. Carefully introducing a variable r for the smaller root and using both the sum and product conditions is the safest method.

Final Answer:
Therefore, the correct relationship between A, B and C is 10B^2 = 49 C A.