If the quadratic equations x^2 - 7x + 3c = 0 | vedclass

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(D) Let $\alpha$ be the common root of the equations $x^2 - 7x + 3c = 0$ and $x^2 + x - 5c = 0$.Then,$\alpha^2 - 7\alpha + 3c = 0$ and $\alpha^2 + \al

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positive for all $x \in R$

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(D) Let $\alpha$ be the common root of the equations $x^2 - 7x + 3c = 0$ and $x^2 + x - 5c = 0$.Then,$\alpha^2 - 7\alpha + 3c = 0$ and $\alpha^2 + \alpha - 5c = 0$.Subtracting the two equations: $(\alpha^2 - 7\alpha + 3c) - (\alpha^2 + \alpha - 5c) = 0$ $\Rightarrow -8\alpha + 8c = 0$ $\Rightarrow \alpha = c$.Substituting $\alpha = c$ into the first equation: $c^2 - 7c + 3c = 0$ $\Rightarrow c^2 - 4c = 0$ $\Rightarrow c(c - 4) = 0$.Since $c$ is non-zero,$c = 4$.The expression becomes $x^2 - 3x + 4$.The discriminant $D = (-3)^2 - 4(1)(4) = 9 - 16 = -7$.Since the coefficient of $x^2$ is $1 > 0$ and $D < 0$,the expression $x^2 - 3x + 4$ is always positive for all $x \in R$.

If the quadratic equations $x^2 - 7x + 3c = 0$ and $x^2 + x - 5c = 0$ have a common root,then for a non-zero real value of $c$,the sign of the expression $x^2 - 3x + c$ is:

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