Solve the equation by completing the square. $$ x^{2}-\frac{5}{2} x-\frac{39}{16}=0 $$ Determine the appropriate process for rearranging the equation before completing the square to obtain an equation of the form $$ (x+k)^{2}= $$ d. Select the correct choice below and fill in the answer box(es) within your choice. (Type an integer or a simplified fraction.) A. $$ x^{2}-\frac{5}{2} x-\frac{39}{16}=0 \Rightarrow x^{2}-\frac{5}{2} x=\frac{39}{16} \Rightarrow x^{2}-\frac{5}{2} x+\square=\frac{39}{16}+\square $$ B. $$ x^{2}-\frac{5}{2} x-\frac{39}{16}=0 \Rightarrow x^{2}-\frac{5}{2} x=\frac{39}{16} \Rightarrow x^{2}-\frac{5}{2} x+\square=\frac{39}{16} $$ C. $$ x^{2}-\frac{5}{2} x-\frac{39}{16}=0 \Rightarrow \frac{2}{5} x^{2}-x=\frac{39}{40} \Rightarrow \frac{2}{5} x^{2}-x+\square=\frac{39}{40}+\square $$ D. $$ x^{2}-\frac{5}{2} x-\frac{39}{16}=0 \Rightarrow \frac{2}{5} x^{2}-x=\frac{39}{40} \Rightarrow \frac{2}{5} x^{2}-x+\square=\frac{39}{40} $$

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UpStudy AI · Accepted Answer

To solve the equation $$ x^{2} - \frac{5}{2}x - \frac{39}{16} = 0 $$ by completing the square, we need to rearrange the equation into a form that's suitable for this method.

**Completed Choice:**

**A.**
$$
x^{2} - \frac{5}{2}x - \frac{39}{16} = 0 \Rightarrow x^{2} - \frac{5}{2}x = \frac{39}{16} \Rightarrow x^{2} - \frac{5}{2}x + \frac{25}{16} = \frac{39}{16} + \frac{25}{16}
$$

**Explanation:**

1. **Isolate the Constant Term:**
   Start by moving the constant term to the right side of the equation:
   $$
   x^{2} - \frac{5}{2}x = \frac{39}{16}
   $$

2. **Determine the Term to Complete the Square:**
   To complete the square, take half of the coefficient of $$ x $$ (which is $$ -\frac{5}{2} $$), square it, and add this value to both sides:
   $$
   \left( \frac{-\frac{5}{2}}{2} \right)^2 = \left( -\frac{5}{4} \right)^2 = \frac{25}{16}
   $$
   
3. **Add the Square Term:**
   Add $$ \frac{25}{16} $$ to both sides to maintain the equality:
   $$
   x^{2} - \frac{5}{2}x + \frac{25}{16} = \frac{39}{16} + \frac{25}{16}
   $$

This rearrangement allows you to write the left side as a perfect square:
$$
\left( x - \frac{5}{4} \right)^2 = \frac{64}{16} = 4
$$

From here, you can solve for $$ x $$ by taking the square root of both sides and isolating $$ x $$.

**Final Answer:**  
**A.**  
$$
x^{2}-\frac{5}{2} x-\frac{39}{16}=0 \Rightarrow x^{2}-\frac{5}{2} x=\frac{39}{16} \Rightarrow x^{2}-\frac{5}{2} x+\frac{25}{16}=\frac{39}{16}+\frac{25}{16}
$$
**A.** $$ x^{2} - \frac{5}{2}x - \frac{39}{16} = 0 \Rightarrow x^{2} - \frac{5}{2}x = \frac{39}{16} \Rightarrow x^{2} - \frac{5}{2}x + \frac{25}{16} = \frac{39}{16} + \frac{25}{16} $$

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