Understanding the Expression
The expression we are working with is x² - 12x + 36. This is a quadratic expression, which can often be factored using algebraic identities.
Recognizing the Pattern
This expression can be recognized as a perfect square trinomial. A perfect square trinomial takes the form:
- (a - b)² = a² - 2ab + b²
In our case, we can identify:
- a = x
- b = 6 (since 6² = 36)
Applying the Identity
Now, we apply the identity to factor the expression:
1. Identify 2ab:
- 2ab = 2 * x * 6 = 12x
2. Rewrite the expression:
- x² - 12x + 36 can be rewritten as:
- x² - 2(6)(x) + 6²
3. Factor using the perfect square identity:
- Thus, we can express it as:
- (x - 6)²
Final Factorization
The expression x² - 12x + 36 can be completely factored as:
- (x - 6)²
Conclusion
In summary, recognizing the structure of the expression as a perfect square trinomial allows us to factor it efficiently. The final factored form is (x - 6)², which demonstrates that x = 6 is a double root of the original quadratic equation. This method not only simplifies calculations but also enhances understanding of polynomial functions.