Which of the following equations has the same solutions as the equation `40-6x=x ^(2) -y` ?

Options

A. `y = (x -6) ^(2) -40`
B. `y = (x -6) ^(2) + 40`
C. `y = (x +3) ^(2) -49`
D. `y = (x +3) ^(2) + 49`

Explanation

To solve the equation \( 40 - 6x = x^2 - y \) and find an equivalent equation that has the same solutions, we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ 40 - 6x = x^2 - y \]
We want to express \( y \) in terms of \( x \). Rearranging gives:
\[ y = x^2 - 40 + 6x \]

### Step 2: Rearranging to Standard Form
Now, we can rearrange the equation to standard form:
\[ y = x^2 + 6x - 40 \]

### Step 3: Completing the Square
Next, we will complete the square for the quadratic expression on the right side. The coefficient of \( x \) is \( 6 \). We take half of this coefficient, square it, and add and subtract it inside the equation:
\[ \text{Half of } 6 = 3 \quad \text{and } 3^2 = 9 \]
So, we add and subtract \( 9 \):
\[ y = (x^2 + 6x + 9) - 9 - 40 \]
This simplifies to:
\[ y = (x + 3)^2 - 49 \]

### Step 4: Final Rearrangement
Now, we can express this in a different form:
\[ y + 49 = (x + 3)^2 \]

### Conclusion
The equation \( y + 49 = (x + 3)^2 \) has the same solutions as the original equation \( 40 - 6x = x^2 - y \).