If alpha, beta are the roots of {x^2} + px + | vedclass

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(C) Given that $\alpha$ and $\beta$ are the roots of ${x^2} + px + q = 0$,we have $\alpha + \beta = -p$ and $\alpha \beta = q$.Given that $\alpha + h$

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${p^2} - 4q = {r^2} - 4s$

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(C) Given that $\alpha$ and $\beta$ are the roots of ${x^2} + px + q = 0$,we have $\alpha + \beta = -p$ and $\alpha \beta = q$.Given that $\alpha + h$ and $\beta + h$ are the roots of ${x^2} + rx + s = 0$,we have $(\alpha + h) + (\beta + h) = -r$ and $(\alpha + h)(\beta + h) = s$.From the first equation: $(\alpha + \beta) + 2h = -r \Rightarrow -p + 2h = -r \Rightarrow h = \frac{p - r}{2} \dots (i)$.From the second equation: $\alpha \beta + h(\alpha + \beta) + h^2 = s$.Substituting the values of $\alpha + \beta$,$\alpha \beta$,and $h$: $q + h(-p) + h^2 = s$.$q + \left( \frac{p - r}{2} \right)(-p) + \left( \frac{p - r}{2} \right)^2 = s$.$q - \frac{p^2 - pr}{2} + \frac{p^2 + r^2 - 2pr}{4} = s$.Multiplying by $4$: $4q - 2p^2 + 2pr + p^2 + r^2 - 2pr = 4s$.$4q - p^2 + r^2 = 4s \Rightarrow r^2 - 4s = p^2 - 4q$.

If $\alpha, \beta$ are the roots of ${x^2} + px + q = 0$ and $\alpha + h, \beta + h$ are the roots of ${x^2} + rx + s = 0$,then

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