If ax^2 + bx + c = 0 and bx^2 + cx + a = 0 ha | vedclass

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(C) Let $\alpha$ be the common root of the equations $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$.Then $a\alpha^2 + b\alpha + c = 0$ and $b\alpha^2 + c

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$3$

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(C) Let $\alpha$ be the common root of the equations $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$.Then $a\alpha^2 + b\alpha + c = 0$ and $b\alpha^2 + c\alpha + a = 0$.Using the method of cross-multiplication:$\frac{\alpha^2}{b(a) - c(c)} = \frac{\alpha}{c(b) - a(a)} = \frac{1}{a(c) - b(b)}$$\frac{\alpha^2}{ab - c^2} = \frac{\alpha}{bc - a^2} = \frac{1}{ac - b^2}$From the first two ratios,$\alpha = \frac{bc - a^2}{ab - c^2}$.From the last two ratios,$\alpha = \frac{ac - b^2}{bc - a^2}$.Equating the two expressions for $\alpha$:$(bc - a^2)^2 = (ab - c^2)(ac - b^2)$$b^2c^2 + a^4 - 2a^2bc = a^2bc - ab^3 - c^3a + b^2c^2$$a^4 + ab^3 + ac^3 = 3a^2bc$Since $a 
eq 0$,dividing by $a$ gives $a^3 + b^3 + c^3 = 3abc$.Therefore,$\frac{a^3 + b^3 + c^3}{abc} = 3$.

If $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root,where $a \neq 0$,then $\frac{a^3 + b^3 + c^3}{abc} = $

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