If one root of ax^2 + bx + c = 0 is the squar | vedclass

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Question

(A) Let the roots be $\alpha$ and $\alpha^2$.From the relation between roots and coefficients:Sum of roots: $\alpha + \alpha^2 = -\frac{b}{a}$ ..... $

Vedclass · Accepted Answer

$3abc$

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(A) Let the roots be $\alpha$ and $\alpha^2$.From the relation between roots and coefficients:Sum of roots: $\alpha + \alpha^2 = -\frac{b}{a}$ ..... $(i)$Product of roots: $\alpha \cdot \alpha^2 = \alpha^3 = \frac{c}{a}$ ..... $(ii)$Cubing equation $(i)$ on both sides:$(\alpha + \alpha^2)^3 = (-\frac{b}{a})^3$$\alpha^3 + (\alpha^2)^3 + 3\alpha \cdot \alpha^2(\alpha + \alpha^2) = -\frac{b^3}{a^3}$Substitute $\alpha^3 = \frac{c}{a}$ and $\alpha + \alpha^2 = -\frac{b}{a}$:$\frac{c}{a} + (\frac{c}{a})^2 + 3(\frac{c}{a})(-\frac{b}{a}) = -\frac{b^3}{a^3}$$\frac{c}{a} + \frac{c^2}{a^2} - \frac{3bc}{a^2} = -\frac{b^3}{a^3}$Multiply the entire equation by $a^3$:$a^2c + ac^2 - 3abc = -b^3$Rearranging the terms,we get:$b^3 + a^2c + ac^2 = 3abc$.

If one root of $ax^2 + bx + c = 0$ is the square of the other,then the value of $b^3 + ac^2 + a^2c$ is

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