Theory of equations

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$\mathrm{If} {\mathrm{x}}^{3}+3{\mathrm{x}}^{2}-9\mathrm{x}+\mathrm{c}$ is of the form ${\left(\mathrm{x}-\mathrm{\alpha }\right)}^{2}\left(\mathrm{x}-\mathrm{\beta }\right)$ then the positive value of c is ____

Moderate
Solution

$\begin{array}{l}\mathrm{Here} \mathrm{x}=\mathrm{\alpha } \mathrm{isarepeatedrootoftheequation} \mathrm{f}\left(\mathrm{x}\right)=0.\\ \mathrm{Hence}, \mathrm{x}=\mathrm{\alpha } \mathrm{isalsoarootoftheequation} \mathrm{f}\text{'}\left(\mathrm{x}\right)=0, \mathrm{i}.\mathrm{e}.,\\ 3{\mathrm{x}}^{2}+6\mathrm{x}-9=0\\ \mathrm{or} {\mathrm{x}}^{2}+2\mathrm{x}-3=0 \mathrm{or} \left(\mathrm{x}+3\right)\left(\mathrm{x}-1\right)=0\\ \mathrm{has} \mathrm{the} \mathrm{root} \mathrm{\alpha } \mathrm{once} \mathrm{which} \mathrm{can} \mathrm{be} \mathrm{eitheer} -3 \mathrm{or} 1.\\ \mathrm{If} \mathrm{\alpha }=1, \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)=0 \mathrm{gives} \mathrm{c}-5=0 \mathrm{or} \mathrm{c}=5.\\ \mathrm{If} \mathrm{\alpha }=-3, \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)=0 \mathrm{gives} -27+27+27+\mathrm{c}=0\\ \therefore \mathrm{c}=-27\end{array}$

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Let three quadratic equations ${\mathrm{ax}}^{2}-2\mathrm{bx}+\mathrm{c}=0,{\mathrm{bx}}^{2}-2\mathrm{cx}+\mathrm{a}=0$ and ${\mathrm{cx}}^{2}-2\mathrm{ax}+\mathrm{b}=0$, all have only positive roots. Then which of these are always true?