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Question

(C) From the given equations:$\alpha + \beta = -b$ .... $(1)$ and $\alpha\beta = ac$ .... $(2)$$\alpha + \gamma = -a$ .... $(3)$ and $\alpha\gamma = b

Vedclass · Accepted Answer

$c, a, b$

Vedclass · Answer

(C) From the given equations:$\alpha + \beta = -b$ .... $(1)$ and $\alpha\beta = ac$ .... $(2)$$\alpha + \gamma = -a$ .... $(3)$ and $\alpha\gamma = bc$ .... $(4)$Subtracting $(3)$ from $(1)$ gives: $\beta - \gamma = a - b$ .... $(5)$Dividing $(2)$ by $(4)$ gives: $\frac{\beta}{\gamma} = \frac{a}{b} \implies \beta = \frac{a\gamma}{b}$ .... $(6)$Substituting $\beta$ from $(6)$ into $(5)$:$\frac{a\gamma}{b} - \gamma = a - b$$\gamma \left( \frac{a - b}{b} ight) = (a - b)$Assuming $a 
eq b$,we get $\gamma = b$.Substituting $\gamma = b$ into $(4)$: $\alpha b = bc \implies \alpha = c$.Substituting $\alpha = c$ into $(2)$: $c\beta = ac \implies \beta = a$.Thus,the values are $\alpha = c, \beta = a, \gamma = b$.

If the roots of the equation $x^2 + bx + ac = 0$ are $\alpha, \beta$ and the roots of the equation $x^2 + ax + bc = 0$ are $\alpha, \gamma$,then what are the values of $\alpha, \beta, \gamma$ respectively?

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