If every pair of the equations x^2 + px + qr | vedclass

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(A) Let the common roots of the pairs of equations be $\alpha$,$\beta$,and $\gamma$ respectively.For the first pair $(x^2 + px + qr = 0)$ and $(x^2 +

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$\frac{-(p + q + r)}{2}$

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(A) Let the common roots of the pairs of equations be $\alpha$,$\beta$,and $\gamma$ respectively.For the first pair $(x^2 + px + qr = 0)$ and $(x^2 + qx + rp = 0)$,let the common root be $\alpha$.For the second pair $(x^2 + qx + rp = 0)$ and $(x^2 + rx + pq = 0)$,let the common root be $\beta$.For the third pair $(x^2 + rx + pq = 0)$ and $(x^2 + px + qr = 0)$,let the common root be $\gamma$.From the properties of roots of quadratic equations,for each equation,the sum of roots is given by the coefficient of $x$ with a negative sign.Thus,$\alpha + \beta = -p$,$\beta + \gamma = -q$,and $\gamma + \alpha = -r$.Adding these three equations,we get $2(\alpha + \beta + \gamma) = -(p + q + r)$.Therefore,the sum of the three common roots is $\alpha + \beta + \gamma = \frac{-(p + q + r)}{2}$.

If every pair of the equations $x^2 + px + qr = 0$,$x^2 + qx + rp = 0$,and $x^2 + rx + pq = 0$ have a common root,then the sum of the three common roots is:

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