If roots of the equation ax^2 + bx + c = 0 ar | vedclass

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Question

(C) For the equation $ax^2 + bx + c = 0$,the roots are $r_1 = \alpha - \beta$ and $r_2 = \gamma - \delta$. The difference of the roots is $|r_1 - r_2|

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$\sqrt{\frac{D_1}{D_2}}$

Vedclass · Answer

(C) For the equation $ax^2 + bx + c = 0$,the roots are $r_1 = \alpha - \beta$ and $r_2 = \gamma - \delta$. The difference of the roots is $|r_1 - r_2| = |(\alpha - \beta) - (\gamma - \delta)| = |\alpha - \beta - \gamma + \delta| = \frac{\sqrt{D_1}}{|a|}$.For the equation $Ax^2 + Bx + C = 0$,the roots are $R_1 = \alpha + \delta$ and $R_2 = \beta + \gamma$. The difference of the roots is $|R_1 - R_2| = |(\alpha + \delta) - (\beta + \gamma)| = |\alpha - \beta + \delta - \gamma| = \frac{\sqrt{D_2}}{|A|}$.Since $|\alpha - \beta - \gamma + \delta| = |\alpha - \beta + \delta - \gamma|$,we have $\frac{\sqrt{D_1}}{|a|} = \frac{\sqrt{D_2}}{|A|}$.Therefore,$\left| \frac{a}{A} \right| = \sqrt{\frac{D_1}{D_2}}$.

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