Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 25c^{2} - 60cd + 36d^{2} = (5c - 6d)^{2} \)
Solution
To factor the quadratic expression \( 25c^{2} - 60cd + 36d^{2} \), we can follow these steps:
1. **Identify Perfect Squares:**
- \( 25c^{2} \) is a perfect square and can be written as \( (5c)^{2} \).
- \( 36d^{2} \) is also a perfect square and can be written as \( (6d)^{2} \).
2. **Check the Middle Term:**
- The middle term is \(-60cd\).
- In a perfect square trinomial of the form \( (a - b)^{2} = a^{2} - 2ab + b^{2} \), the middle term should be \( -2ab \).
- Here, \( a = 5c \) and \( b = 6d \), so \( -2ab = -2 \times 5c \times 6d = -60cd \), which matches the middle term.
3. **Write the Factored Form:**
- Since all conditions for a perfect square trinomial are met, the expression factors to:
\[
(5c - 6d)^{2}
\]
**Final Answer:**
\[
25c^{2} - 60cd + 36d^{2} = (5c - 6d)^{2}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To factor the quadratic expression \( 25c^2 - 60cd + 36d^2 \), you can look for two binomials whose product gives the original expression. The expression can be recognized as a perfect square trinomial, specifically \( (5c - 6d)^2 \). To verify, you can expand this binomial and ensure it results in the original expression. Additionally, this quadratic form is commonly encountered in algebra, especially when solving problems relating to area, geometry, or optimization. Recognizing patterns in factoring can significantly speed up problem-solving and enhance your understanding of polynomial relationships!
