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(C) For a quadratic equation of the form $ax^{2} + bx + c = 0$,the roots are equal and real if the discriminant $D = b^{2} - 4ac = 0$.Given the equati

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$51$

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(C) For a quadratic equation of the form $ax^{2} + bx + c = 0$,the roots are equal and real if the discriminant $D = b^{2} - 4ac = 0$.Given the equation: $16x^{2} - 40x + \frac{k-1}{2} = 0$.Here,$a = 16$,$b = -40$,and $c = \frac{k-1}{2}$.Substituting these values into the discriminant formula:$(-40)^{2} - 4(16)\left(\frac{k-1}{2}\right) = 0$$1600 - 64\left(\frac{k-1}{2}\right) = 0$$1600 - 32(k-1) = 0$$1600 - 32k + 32 = 0$$1632 = 32k$$k = \frac{1632}{32} = 51$.Thus,the value of $k$ is $51$.

If the following quadratic equation has two equal and real roots,then find the value of $k$: $16x^{2} - 40x + \frac{k-1}{2} = 0$.

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