Question

Let a, b, c be real numbers with $a ≠ 0$ and let $α, β$ be the roots of the equation $ax²+bx+c=0$, then $a³x²+abcx+c³=0$ has roots

• A. $\alpha^2 \beta, \beta^2 \alpha$
• B. $\alpha, \beta^2$
• C. $\alpha^2 \beta, \beta \alpha$
• D. $\alpha^3 \beta, \beta^3 \alpha$

Hint:

Let a, b, c be real numbers with $a ≠ 0$ and let $α, β$ be the roots of the equation $ax+bx+c=0$, then $ax+abcx+c=0$ has roots

Answer 1

The Correct Option is A

Step 1: Concept
Use the relationship between roots and coefficients: $\alpha+\beta = -b/a$ and $\alpha\beta = c/a$.
Step 2: Analysis
Divide the second equation $a^3x^2+abcx+c^3=0$ by $c^2$ to get $a(\frac{ax}{c})^2 + b(\frac{ax}{c}) + c = 0$.
Step 3: Evaluation
The term $(\frac{ax}{c})$ must be equal to the roots of the original equation, $\alpha$ and $\beta$. Thus, $x = \frac{c}{a}\alpha$ and $x = \frac{c}{a}\beta$.
Step 4: Conclusion
Substituting $\frac{c}{a} = \alpha\beta$, the roots are $x = (\alpha\beta)\alpha = \alpha^2\beta$ and $x = (\alpha\beta)\beta = \alpha\beta^2$.
Final Answer: (a)