Question
6) \( b^{2}+16 b+64 \)
Ask by Turnbull Horton. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( b^{2} + 16b + 64 = (b + 8)^{2} \)
Solution
To factor the quadratic expression \( b^{2} + 16b + 64 \), follow these steps:
1. **Identify the coefficients:**
- Quadratic term coefficient (\( a \)): 1
- Linear term coefficient (\( b \)): 16
- Constant term (\( c \)): 64
2. **Find two numbers that multiply to \( a \times c = 1 \times 64 = 64 \) and add up to \( b = 16 \).**
- The numbers **8 and 8** satisfy these conditions because:
- \( 8 \times 8 = 64 \)
- \( 8 + 8 = 16 \)
3. **Rewrite the quadratic expression using these numbers:**
\[
b^{2} + 8b + 8b + 64
\]
4. **Factor by grouping:**
\[
(b^{2} + 8b) + (8b + 64) = b(b + 8) + 8(b + 8)
\]
5. **Factor out the common binomial factor \((b + 8)\):**
\[
(b + 8)(b + 8) = (b + 8)^{2}
\]
**Final Factored Form:**
\[
b^{2} + 16b + 64 = (b + 8)^{2}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
Did you know that the expression \( b^{2}+16b+64 \) is actually a perfect square? It can be factored as \( (b+8)^{2} \). This means it represents a quadratic function that opens upwards with its vertex at the point (-8, 0). It’s like having a cozy little mountain on a graph! In terms of real-world application, perfect square trinomials like this one pop up in various fields, particularly in physics and engineering where quadratic equations describe trajectories, optimization problems, and structures. So next time you see a quadratic form, think of it as a little gem hiding a useful application!
