The value of c for which |{alpha ^2} - {beta | vedclass

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(C) Given the quadratic equation $2x^2 + 7x + c = 0$.For this equation,the sum of the roots $\alpha + \beta = -\frac{7}{2}$ and the product of the roo

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$6$

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(C) Given the quadratic equation $2x^2 + 7x + c = 0$.For this equation,the sum of the roots $\alpha + \beta = -\frac{7}{2}$ and the product of the roots $\alpha \beta = \frac{c}{2}$.We are given $|\alpha^2 - \beta^2| = \frac{7}{4}$,which implies $(\alpha + \beta)(\alpha - \beta) = \pm \frac{7}{4}$.We know that $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha \beta = (-\frac{7}{2})^2 - 4(\frac{c}{2}) = \frac{49}{4} - 2c$.Thus,$|\alpha - \beta| = \sqrt{\frac{49 - 8c}{4}} = \frac{\sqrt{49 - 8c}}{2}$.Substituting these into the equation $(\alpha + \beta)(\alpha - \beta) = \pm \frac{7}{4}$:$(-\frac{7}{2}) \cdot (\pm \frac{\sqrt{49 - 8c}}{2}) = \pm \frac{7}{4}$.This simplifies to $\frac{7}{4} \sqrt{49 - 8c} = \frac{7}{4}$.Therefore,$\sqrt{49 - 8c} = 1$.Squaring both sides,$49 - 8c = 1$,which gives $8c = 48$,so $c = 6$.

The value of $c$ for which $|{\alpha ^2} - {\beta ^2}| = \frac{7}{4}$,where $\alpha$ and $\beta$ are the roots of $2{x^2} + 7x + c = 0$,is

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