To solve the equation \( x^2 + y^2 + 16 + 2y = -36 \), let's follow these steps:

1. Simplify and rearrange the equation to standard form.
2. Complete the square for the \(y\)-terms to convert the equation into a recognizable form of a circle.

### Step-by-Step Solution

1. **Rearrange the equation:**

   \( x^2 + y^2 + 2y + 16 = -36 \)

   Subtract 16 from both sides to simplify:

   \( x^2 + y^2 + 2y = -52 \)

2. **Complete the square for the \(y\)-terms:**

   The equation is \( x^2 + y^2 + 2y = -52 \). Let's complete the square for the \(y\)-terms.

   For the term \( y^2 + 2y \), take half of the coefficient of \(y\) (which is 2), square it, and add and subtract this value inside the equation:

   \( y^2 + 2y + 1 - 1 \)

   This simplifies to:

   \( (y + 1)^2 - 1 \)



Learn how to solve the equation x² + y² + 16 + 2y = -36 with a detailed step-by-step solution. This problem involves algebraic concepts like quadratic equations and completing the square. Suitable for students in Grades 9-12.

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