Question
Consider the equation: \( x^{2}+12 x+35=0 \) A) First, use the "completing the square" process to write this equation in the form \( (x+D)^{2}=E \) and enter your results below. \( x^{2}+12 x+35=0 \) is equivalent to: Preview left side of eqn:
Ask by Dickson Harmon. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x^{2} + 12x + 35 = 0 \) is equivalent to \( (x + 6)^2 = 1 \).
Solution
To complete the square for the equation \( x^{2} + 12x + 35 = 0 \), follow these steps:
1. **Identify the coefficient of \( x \):**
- The coefficient of \( x \) is **12**.
2. **Divide this coefficient by 2 and square it:**
- \( \frac{12}{2} = 6 \)
- \( 6^2 = 36 \)
3. **Rewrite the equation by adding and subtracting this square inside the equation:**
\[
x^{2} + 12x + 36 - 36 + 35 = 0
\]
\[
(x + 6)^2 - 1 = 0
\]
4. **Express the equation in the completed square form:**
\[
(x + 6)^2 = 1
\]
**Final Completed Square Form:**
\[
(x + 6)^2 = 1
\]
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Simplify this solution Extra Insights
To complete the square for the equation \( x^{2}+12 x+35=0 \), we start by focusing on the \( x^{2} + 12x \) part. We take half of the coefficient of \( x \), which is 12, so half of it is 6, and then we square it, giving us 36. Now, we can rewrite the left-hand side: \[ x^{2} + 12x + 36 - 36 + 35 = 0 \] which simplifies to \[ (x + 6)^{2} - 1 = 0. \] In this form, we can isolate the square: \[ (x + 6)^{2} = 1. \]
