To factorize the expression \( 81a^2 - 90ab + 25b^2 \), let's recognize that it can be written in the form of a perfect square trinomial.

1. **Identify a pattern**: Notice that \(81a^2\) and \(25b^2\) are both perfect squares:
   - \( 81a^2 = (9a)^2 \)
   - \( 25b^2 = (5b)^2 \)

2. **Check the middle term**: The middle term is \(-90ab\), which should match the form \(2 \cdot 9a \cdot 5b\):
   - \( 2 \cdot 9a \cdot 5b = 90ab \)
   - Since the middle term is \(-90ab\), it fits the form \( -2 \cdot 9a \cdot 5b \).

3. **Write the expression as a perfect square**:
   \[
   81a^2 - 90ab + 25b^2 = (9a - 5b)^2
   \]

So, the factorized form of \( 81a^2 - 90ab + 25b^2 \) is:
\[
(9a - 5b)^2
\]

Would you like more details or have any questions?

Here are 8 related questions to expand on this topic:
1. What are perfect square trinomials?
2. How do you identify if a quadratic expression is a perfect square trinomial?
3. Can you provide an example of factorizing a different perfect square trinomial?
4. How can you factorize non-perfect square trinomials?
5. What are the steps to factorize a general quadratic expression \( ax^2 + bx + c \)?
6. How do you use the quadratic formula to solve quadratic equations?
7. What is the difference between factorizing and solving a quadratic equation?
8. How do you handle trinomials with negative coefficients or mixed signs?

**Tip**: Always check the middle term when factorizing a trinomial to see if it fits the form of a perfect square trinomial, which is \(a^2 \pm 2ab + b^2 = (a \pm b)^2\).

Learn how to factorize the expression 81a^2 - 90ab + 25b^2 using the perfect square trinomial method. This detailed solution covers identifying perfect squares, checking middle terms, and writing expressions as perfect squares. Suitable for high school students studying algebra and quadratic equations.

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