Let alpha, beta, gamma (alpha

Question

(C) Given that $\alpha, \beta, \gamma$ are the roots of $ax^3+bx^2+cx+d=0$ and $\beta^2=\alpha \gamma$,it implies that $\alpha, \beta, \gamma$ are in

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$v^2=uw$

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(C) Given that $\alpha, \beta, \gamma$ are the roots of $ax^3+bx^2+cx+d=0$ and $\beta^2=\alpha \gamma$,it implies that $\alpha, \beta, \gamma$ are in a Geometric Progression ($G$.$P$.).Now,consider the equation $ak^3x^3+bk^2x^2+ckx+d=0$. This can be rewritten as $a(kx)^3+b(kx)^2+c(kx)+d=0$.Let $y = kx$. Then the equation becomes $ay^3+by^2+cy+d=0$,which has roots $\alpha, \beta, \gamma$.Therefore,$kx = \alpha, kx = \beta, kx = \gamma$,which gives $x = \frac{\alpha}{k}, x = \frac{\beta}{k}, x = \frac{\gamma}{k}$.Thus,the roots $u, v, w$ are $\frac{\alpha}{k}, \frac{\beta}{k}, \frac{\gamma}{k}$.Since $\alpha, \beta, \gamma$ are in $G$.$P$.,their scaled values $\frac{\alpha}{k}, \frac{\beta}{k}, \frac{\gamma}{k}$ are also in $G$.$P$.Therefore,$u, v, w$ are in $G$.$P$.,which implies $v^2=uw$.

Let $\alpha, \beta, \gamma$ $(\alpha < \beta < \gamma)$ be roots of $ax^3+bx^2+cx+d=0$ and $u, v, w$ $(u < v < w)$ be roots of $ak^3x^3+bk^2x^2+ckx+d=0$. If $\beta^2=\alpha \gamma$,then:

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