If the quadratic equations 3x^2 + ax + 1 = 0 | vedclass

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(B) Let $\alpha$ be the common root of the equations $3x^2 + ax + 1 = 0$ and $2x^2 + bx + 1 = 0$.Since $\alpha$ is a root,we have:$3\alpha^2 + a\alpha

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$1$

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(B) Let $\alpha$ be the common root of the equations $3x^2 + ax + 1 = 0$ and $2x^2 + bx + 1 = 0$.Since $\alpha$ is a root,we have:$3\alpha^2 + a\alpha + 1 = 0$  --- $(1)$$2\alpha^2 + b\alpha + 1 = 0$  --- $(2)$Subtracting $(2)$ from $(1)$,we get:$(3\alpha^2 + a\alpha + 1) - (2\alpha^2 + b\alpha + 1) = 0$$\alpha^2 + (a - b)\alpha = 0$$\alpha(\alpha + a - b) = 0$Since $\alpha = 0$ does not satisfy the equations (as $1 
eq 0$),we must have $\alpha = b - a$.Substituting $\alpha = b - a$ into equation $(2)$:$2(b - a)^2 + b(b - a) + 1 = 0$$2(b^2 - 2ab + a^2) + b^2 - ab + 1 = 0$$2b^2 - 4ab + 2a^2 + b^2 - ab + 1 = 0$$3b^2 - 5ab + 2a^2 + 1 = 0$Rearranging the terms,we get:$2a^2 + 3b^2 - 5ab = -1$Therefore,$5ab - 2a^2 - 3b^2 = -(-1) = 1$.

If the quadratic equations $3x^2 + ax + 1 = 0$ and $2x^2 + bx + 1 = 0$ have a common root,then what is the value of the expression $5ab - 2a^2 - 3b^2$?

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