Question

The condition that one root of the equation $ax^{2} + bx + c = 0$ may be the square of the other is

• A. $a^{2}c + ac^{2} + b^{3} - 3abc = 0$
• B. $a^{2}c^{2} + ac^{2} + b^{2} - 3abc = 0$
• C. $ac^{2} + ac - b^{3} - 3abc = 0$
• D. $a^{2}c + ac^{2} - b^{3} - 3abc = 0$

Hint:

Vieta's formulas: sum of roots $= -b/a$, product $= c/a$. For conditions on roots, set up and eliminate the root variable using these relations.

Answer 1

The Correct Option is D

Step 1: Conceptual Understanding:
Let the roots be $\alpha$ and $\alpha^2$. Use Vieta's formulas and eliminate $\alpha$.
Step 2: Explanation in Detail:
$\alpha + \alpha^2 = -\dfrac{b}{a}$ and $\alpha^3 = \dfrac{c}{a}$.
Cubing the sum relation and using the product gives, after elimination of $\alpha$:
$a^2c + ac^2 - b^3 - 3abc = 0$.
Step 3: Therefore, Stating the Final Answer
$a^2c + ac^2 - b^3 - 3abc = 0$.