Question
What is the exact solution to the equation \( (x+4)^{2}-16=-4 \) ? \( x=2 \sqrt{3} \) \( x=4 \sqrt{3} \) \( x=2 \pm 1 \sqrt{3} \) \( x=-2 \pm 1 \sqrt{3} \) \( x=6 \sqrt{3} \) \( x= \pm 4 \) \( x=-4 \pm 2 \sqrt{3} \) \( x=4 \pm 2 \sqrt{3} \) \( x= \pm 2 \sqrt{3} \) \( x= \pm 2 \)
Ask by Alexander Hall. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The exact solutions are \( x = -4 + 2\sqrt{3} \) and \( x = -4 - 2\sqrt{3} \).
Solution
To solve the equation \( (x + 4)^2 - 16 = -4 \), follow these steps:
1. **Start with the original equation:**
\[
(x + 4)^2 - 16 = -4
\]
2. **Add 16 to both sides to isolate the squared term:**
\[
(x + 4)^2 = 12
\]
3. **Take the square root of both sides:**
\[
x + 4 = \pm \sqrt{12}
\]
Simplify \(\sqrt{12}\):
\[
\sqrt{12} = 2\sqrt{3}
\]
So,
\[
x + 4 = \pm 2\sqrt{3}
\]
4. **Subtract 4 from both sides to solve for \(x\):**
\[
x = -4 \pm 2\sqrt{3}
\]
**Therefore, the exact solutions are:**
\[
x = -4 + 2\sqrt{3} \quad \text{and} \quad x = -4 - 2\sqrt{3}
\]
Looking at the provided options, the correct answer matches:
**\( x = -4 \pm 2\sqrt{3} \)**
**Answer:**
\( x=-4 \pm 2 \sqrt{3} \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To solve the equation \( (x+4)^{2}-16=-4 \), we first simplify it. Rearranging gives us \( (x+4)^{2} = 12 \). Taking the square root of both sides leads to \( x+4 = \pm 2\sqrt{3} \). Solving for \( x \) gives us \( x = -4 \pm 2\sqrt{3} \), which matches one of your options! If we apply this in a real-world context, finding the value of \( x \) could represent a physical phenomenon, like the position of an object in a parabolic path. The ability to bridge algebra and geometry showcases the practical power of math in analyzing movement, such as a ball thrown from a height or the trajectory of a rocket. Math isn't just numbers—it's a tool to explore and understand the world around us!
