For the quadratic equation \( ax^2 + bx + c = 0 \) to have equal roots, its discriminant \( \Delta = b^2 - 4ac \) must be zero. For the given equation \( x^2 - 5x + k = 0 \), we have \( a = 1 \), \( b = -5 \), and \( c = k \). The discriminant is therefore:
\(\Delta = (-5)^2 - 4 \cdot 1 \cdot k = 0\)
Simplifying this equation yields:
\(25 - 4k = 0\)
Solving for \( k \) provides:
\(4k = 25\)
\(k = \frac{25}{4}\)
Consequently, the value of \( k \) that guarantees equal roots for the quadratic equation is \(\frac{25}{4}\).