For the quadratic equation \(ax^2 + bx + c = 0\) to have exactly one root, the discriminant must be zero. The discriminant \(\Delta\) is given by \(b^2 - 4ac\).

Here, \(a = 16\), \(b = -24\), and \(c = 10 - k\).

Set the discriminant to zero:

\((-24)^2 - 4 \times 16 \times (10 - k) = 0\)

\(576 - 64(10 - k) = 0\)

\(576 - 640 + 64k = 0\)

\(-64 + 64k = 0\)

\(64k = 64\)

\(k = 1\)

Substitute \(k = 1\) back into the equation:

\(16x^2 - 24x + 9 = 0\)

Factor the quadratic:

\((4x - 3)^2 = 0\)

Thus, \(4x - 3 = 0\)

\(4x = 3\)

\(x = \frac{3}{4}\)