Understanding the Quadratic Expression
To factorize the quadratic expression \(x^2 - 9x + 5\), we need to express it in the form \((x - p)(x - q)\), where \(p\) and \(q\) are the roots of the equation.

Step 1: Identify Coefficients
- The quadratic expression is in the standard form \(ax^2 + bx + c\).
- Here, \(a = 1\), \(b = -9\), and \(c = 5\).

Step 2: Calculate the Discriminant
- The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\).
- Substitute the values:
\(D = (-9)^2 - 4(1)(5)\)
\(D = 81 - 20\)
\(D = 61\)

Step 3: Determine Roots Using the Quadratic Formula
- The roots can be found using the formula:
\(x = \frac{-b \pm \sqrt{D}}{2a}\).
- Substitute the values:
\(x = \frac{-(-9) \pm \sqrt{61}}{2(1)}\)
\(x = \frac{9 \pm \sqrt{61}}{2}\)

Step 4: Write the Factorized Form
- The factorized form of the expression is:
\(x^2 - 9x + 5 = \left(x - \frac{9 + \sqrt{61}}{2}\right)\left(x - \frac{9 - \sqrt{61}}{2}\right)\).

Conclusion
- The expression \(x^2 - 9x + 5\) does not factor nicely into integers but can be expressed with its roots involving irrational numbers.
- Understanding the steps of finding roots is crucial for factorization in quadratic equations.