Problem Statement
To divide 51 into two parts such that their sum is 51 and their product is 560. Let's denote the two parts as \( x \) and \( y \).

Setting Up the Equations
- The sum equation can be represented as:
- \( x + y = 51 \)
- The product equation can be represented as:
- \( x \cdot y = 560 \)

Expressing One Variable in Terms of Another
By rearranging the sum equation, we can express \( y \) in terms of \( x \):
- \( y = 51 - x \)

Substituting into the Product Equation
Now, substitute \( y \) into the product equation:
- \( x \cdot (51 - x) = 560 \)
This simplifies to:
- \( 51x - x^2 = 560 \)
Rearranging gives us a standard quadratic equation:
- \( x^2 - 51x + 560 = 0 \)

Solving the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1 \), \( b = -51 \), and \( c = 560 \).
Calculating the discriminant:
- \( b^2 - 4ac = 51^2 - 4 \cdot 1 \cdot 560 = 2601 - 2240 = 361 \)
Now, substituting into the formula:
- \( x = \frac{51 \pm \sqrt{361}}{2} \)
- \( x = \frac{51 \pm 19}{2} \)
This gives two possible values for \( x \):
- \( x = \frac{70}{2} = 35 \)
- \( x = \frac{32}{2} = 16 \)

Final Parts
Thus, the two parts are:
- \( x = 35 \) and \( y = 16 \)
- Verification:
- Sum: \( 35 + 16 = 51 \)
- Product: \( 35 \cdot 16 = 560 \)
These two numbers satisfy the conditions given in the problem.