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(D) Given the quadratic equation $4x^2 - \sqrt{13}x - 7 = 0$. Comparing this with $ax^2 + bx + c = 0$,we get $a = 4$,$b = -\sqrt{13}$,and $c = -7$. Th

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$\frac{5\sqrt{5}}{4}$

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(D) Given the quadratic equation $4x^2 - \sqrt{13}x - 7 = 0$. Comparing this with $ax^2 + bx + c = 0$,we get $a = 4$,$b = -\sqrt{13}$,and $c = -7$. The sum of roots $\alpha + \beta = -\frac{b}{a} = -\frac{-\sqrt{13}}{4} = \frac{\sqrt{13}}{4}$. The product of roots $\alpha \beta = \frac{c}{a} = -\frac{7}{4}$. We know that $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$. Substituting the values: $(\alpha - \beta)^2 = (\frac{\sqrt{13}}{4})^2 - 4(-\frac{7}{4}) = \frac{13}{16} + 7 = \frac{13 + 112}{16} = \frac{125}{16}$. Therefore,$|\alpha - \beta| = \sqrt{\frac{125}{16}} = \frac{5\sqrt{5}}{4}$.

If $\alpha$ and $\beta$ are the roots of the equation $4x^2 - \sqrt{13}x - 7 = 0$,then what is the value of $|\alpha - \beta|$?

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