A line segment AB is 8 cm in length. AB is produced to P such that `BP^(2)=AB.AP,` find the length of BP.

Answer

To solve the problem, we need to find the length of segment BP given that AB is 8 cm and \( BP^2 = AB \cdot AP \).

### Step-by-Step Solution:

1. **Identify Given Information:**
- Length of segment AB = 8 cm.
- Let the length of segment BP be \( x \).

2. **Express AP in terms of x:**
- Since AB is produced to point P, the length of AP can be expressed as:
\[ AP = AB + BP = 8 + x \]

3. **Set Up the Equation:**
- According to the problem, we have:
\[ BP^2 = AB \cdot AP \]
- Substituting the known values, we get:
\[ x^2 = 8 \cdot (8 + x) \]

4. **Expand the Right Side:**
- Expanding the equation gives:
\[ x^2 = 64 + 8x \]

5. **Rearrange the Equation:**
- Move all terms to one side to form a quadratic equation:
\[ x^2 - 8x - 64 = 0 \]

6. **Identify Coefficients:**
- In the quadratic equation \( ax^2 + bx + c = 0 \), we have:
- \( a = 1 \)
- \( b = -8 \)
- \( c = -64 \)

7. **Use the Quadratic Formula:**
- The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Plugging in the values:
\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-64)}}{2 \cdot 1} \]

8. **Calculate the Discriminant:**
- Calculate \( b^2 - 4ac \):
\[ (-8)^2 = 64 \]
\[ -4 \cdot 1 \cdot (-64) = 256 \]
\[ b^2 - 4ac = 64 + 256 = 320 \]

9. **Substitute Back into the Formula:**
- Now substitute back:
\[ x = \frac{8 \pm \sqrt{320}}{2} \]

10. **Simplify \( \sqrt{320} \):**
- \( \sqrt{320} = \sqrt{64 \cdot 5} = 8\sqrt{5} \)
- So, we have:
\[ x = \frac{8 \pm 8\sqrt{5}}{2} \]

11. **Final Simplification:**
- Simplifying gives:
\[ x = 4 \pm 4\sqrt{5} \]

12. **Select the Positive Value:**
- Since length cannot be negative, we take:
\[ x = 4 + 4\sqrt{5} \]

### Conclusion:
The length of BP is:
\[ BP = 4(1 + \sqrt{5}) \text{ cm} \]