Question
Real Tutor Solution
Tutor-Verified Answer
Answer
The missing number to complete the square is 9.
Solution
To complete the square for the quadratic expression \( w^2 + 6w + \square \), follow these steps:
1. **Identify the Coefficient of \( w \):**
The coefficient of \( w \) is **6**.
2. **Take Half of This Coefficient:**
\[
\frac{6}{2} = 3
\]
3. **Square the Result from Step 2:**
\[
3^2 = 9
\]
4. **Insert This Value into the Expression:**
\[
w^2 + 6w + 9
\]
5. **Write the Completed Square:**
The expression \( w^2 + 6w + 9 \) can be written as:
\[
(w + 3)^2
\]
**Final Completed Expression:**
\[
w^2 + 6w + \mathbf{9} = (w + 3)^2
\]
So, the missing number to complete the square is **9**.
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Simplify this solution Bonus Knowledge
The expression \( w^2 + 6w + \square \) hints at completing the square. To make it a perfect square trinomial, we need to find the value of the square. Half of the coefficient of \( w \) (which is 6) is 3, and squaring that gives us 9. Therefore, the expression becomes \( w^2 + 6w + 9 \), which can be factored into \( (w+3)^2 \). Now, if you're ever faced with such an expression, remember to always take half of the linear coefficient, square it, and add it to complete the square. A common mistake is to overlook the need for adding this term, which can lead to an incomplete or incorrect factorization.
