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Question

(C) Let the roots of the equation $6x^3 + 7x^2 - 4x - 2 = 0$ be $\alpha, \beta, \gamma$. If the roots are diminished by $h$,the new roots are $\alpha-

Vedclass · Accepted Answer

$-2/9$

Vedclass · Answer

(C) Let the roots of the equation $6x^3 + 7x^2 - 4x - 2 = 0$ be $\alpha, \beta, \gamma$. If the roots are diminished by $h$,the new roots are $\alpha-h, \beta-h, \gamma-h$. Let $y = x - h$,so $x = y + h$. Substituting $x = y + h$ into the original equation: $6(y+h)^3 + 7(y+h)^2 - 4(y+h) - 2 = 0$. Expanding this,the coefficient of $y^2$ is $6(3h) + 7 = 18h + 7$. For the transformed equation to not contain the $y^2$ term,we set $18h + 7 = 0$,which gives $h = -7/18$. However,the question asks for the condition where the $x$ term (linear term) is missing. Expanding $6(y+h)^3 + 7(y+h)^2 - 4(y+h) - 2 = 0$: $6(y^3 + 3y^2h + 3yh^2 + h^3) + 7(y^2 + 2yh + h^2) - 4(y+h) - 2 = 0$. The coefficient of $y$ is $18h^2 + 14h - 4 = 0$. Dividing by $2$,we get $9h^2 + 7h - 2 = 0$. The product of the roots $h$ for this quadratic equation is $c/a = -2/9$.

After the roots of the equation $6x^3 + 7x^2 - 4x - 2 = 0$ are diminished by $h$,if the transformed equation does not contain the $x^2$ term,then the product of all the possible values of $h$ is

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