To factor the quadratic expression \(x^2 - 13x + 36\), we need to find two numbers that multiply to \(36\) (the constant term) and add to \(-13\) (the coefficient of the linear term).
The two numbers that satisfy these conditions are \(-9\) and \(-4\) because:
\[ -9 \times -4 = 36 \]
and
\[ -9 + -4 = -13. \]
Therefore, we can factor the expression as:
\[ x^2 - 13x + 36 = (x - 9)(x - 4). \]
So, the complete factored form is:
\[ (x - 9)(x - 4). \]