Question
Solve the following equation by the square root method. \( (2 y+8)^{2}=64 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \} \) (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There is no solution.
Ask by Campbell Osborne. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution set is \{-8, 0\}.
Solution
To solve the equation \((2y + 8)^2 = 64\) using the square root method, follow these steps:
1. **Take the square root of both sides:**
\[
\sqrt{(2y + 8)^2} = \sqrt{64}
\]
This simplifies to:
\[
2y + 8 = \pm 8
\]
2. **Solve for \( y \) in both cases:**
- **Case 1:** \( 2y + 8 = 8 \)
\[
2y = 8 - 8
\]
\[
2y = 0
\]
\[
y = 0
\]
- **Case 2:** \( 2y + 8 = -8 \)
\[
2y = -8 - 8
\]
\[
2y = -16
\]
\[
y = -8
\]
3. **Solution Set:**
The solutions are \( y = 0 \) and \( y = -8 \).
**Therefore, the correct choice is:**
**A. The solution set is \{-8, 0\}**
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Simplify this solution Bonus Knowledge
To solve the equation \( (2y + 8)^2 = 64 \) using the square root method, we first take the square root of both sides: \[ 2y + 8 = \pm 8 \] This gives us two cases to solve: 1. \( 2y + 8 = 8 \) 2. \( 2y + 8 = -8 \) For the first case: \[ 2y = 8 - 8 \] \[ 2y = 0 \] \[ y = 0 \] For the second case: \[ 2y = -8 - 8 \] \[ 2y = -16 \] \[ y = -8 \] Thus, the solution set is \( \{ 0, -8 \} \). **Choice A:** The solution set is \( \{ 0, -8 \} \).
