p and q are two roots of the equation x^2+7x+ | vedclass

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(C) Given that $p$ and $q$ are roots of the quadratic equation $x^2+7x+3=0$. To find the quadratic equation whose roots are $\frac{3p}{1-2p}$ and $\fr

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(C) Given that $p$ and $q$ are roots of the quadratic equation $x^2+7x+3=0$. To find the quadratic equation whose roots are $\frac{3p}{1-2p}$ and $\frac{3q}{1-2q}$,let $y = \frac{3x}{1-2x}$. Then $y(1-2x) = 3x$ $\Rightarrow y - 2xy = 3x$ $\Rightarrow y = x(3+2y)$ $\Rightarrow x = \frac{y}{3+2y}$. Since $x$ is a root of $x^2+7x+3=0$,we substitute $x = \frac{y}{3+2y}$ into the equation: $(\frac{y}{3+2y})^2 + 7(\frac{y}{3+2y}) + 3 = 0$. Multiplying by $(3+2y)^2$,we get $y^2 + 7y(3+2y) + 3(3+2y)^2 = 0$. $y^2 + 21y + 14y^2 + 3(9 + 12y + 4y^2) = 0$. $15y^2 + 21y + 27 + 36y + 12y^2 = 0$. $27y^2 + 57y + 27 = 0$. Dividing by $3$,we get $9y^2 + 19y + 9 = 0$. Comparing this with $lx^2+mx+n=0$,we have $l=9, m=19, n=9$. The greatest common divisor of $9, 19, 9$ is $1$. Therefore,$l-m+n = 9 - 19 + 9 = -1$.

$p$ and $q$ are two roots of the equation $x^2+7x+3=0$. If $\frac{3p}{1-2p}$ and $\frac{3q}{1-2q}$ are the roots of $lx^2+mx+n=0$ and the greatest common divisor of $l, m, n$ is $1$,then $l-m+n=$

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