The given equation is a quadratic equation of the form ax² + bx + c = 0, where a = 1, b = -10, and c = k. A quadratic equation has only one solution when its discriminant (b² - 4ac) is equal to zero.
So we set the discriminant to zero:
(-10)² - 4(1)(k) = 0
100 - 4k = 0
4k = 100
k = 25
Therefore, the value of k is $\boxed{25}$.