Question
12. Factor each trinomial completely, if possible. If it is not possible to factor, enter the original polynomial into the answerbox. \( x^{2}-10 x+25 \) Question Help: \( \square \) Video Submit Question
Ask by Barrett Blake. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( (x - 5)^2 \)
Solution
To factor the trinomial \( x^{2} - 10x + 25 \), follow these steps:
1. **Identify the coefficients:**
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -10 \) (coefficient of \( x \))
- \( c = 25 \) (constant term)
2. **Find two numbers that multiply to \( a \cdot c = 1 \cdot 25 = 25 \) and add up to \( b = -10 \):**
- The numbers are **-5** and **-5** because:
- \( -5 \times -5 = 25 \)
- \( -5 + -5 = -10 \)
3. **Write the factored form using these numbers:**
\[
x^{2} - 10x + 25 = (x - 5)(x - 5) = (x - 5)^2
\]
**Final Answer:**
\[
(x - 5)^2
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
The trinomial \( x^{2}-10x+25 \) is a perfect square trinomial. It can be factored as \( (x-5)(x-5) \) or simply written as \( (x-5)^{2} \). Perfect square trinomials occur when the first and last terms are both perfect squares and the middle term is twice the product of their square roots. Factoring polynomials like this is not just useful in math class! In real-world applications, such as physics and engineering, factoring helps simplify complex equations. Engineers often factor to find points of maximum or minimum, which is crucial when designing structures or systems that need to perform optimally under given conditions.
