Solution: Group terms: $ 9(x^2 - 2x) - 16(y^2 + 4y) = 144 $. Complete the square: $ 9[(x - 1)^2 - 1] - 16[(y + 2)^2 - 4] = 144 $. Expand: $ 9(x - 1)^2 - 9 - 16(y + 2)^2 + 64 = 144 $. Simplify: $ 9(x - 1)^2 - 16(y + 2)^2 = 89 $. The center is at $ (1, -2) $. Final answer: $oxed

["Completing the Square: Solving the Equation $ 9(x^2 - 2x) - 16(y^2 + 4y) = 144 $", "Working with quadratic expressions involving two variables can seem complex, but by completing the square, we can transform this equation into a recognizable standard form. Let’s walk through the solution step by step.", "---", "### Step 1: Group the terms", "We begin by grouping $ x $ and $ y $ terms together:\n[\n9(x^2 - 2x) - 16(y^2 + 4y) = 144\n]", "---", "### Step 2: Complete the square", "To complete the square:", "- For $ x^2 - 2x $:\n [\n x^2 - 2x = (x - 1)^2 - 1\n ]", "- For $ y^2 + 4y $:\n [\n y^2 + 4y = (y + 2)^2 - 4\n ]", "Substitute these back into the equation:\n[\n9\left[(x - 1)^2 - 1\right] - 16\left[(y + 2)^2 - 4\right] = 144\n]", "---", "### Step 3: Expand the terms", "Expanding both squared expressions:\n[\n9(x - 1)^2 - 9 - 16(y + 2)^2 + 64 = 144\n]", "Combine constants on the left:\n[\n9(x - 1)^2 - 16(y + 2)^2 + 55 = 144\n]", "---", "### Step 4: Simplify to standard form", "Move the constant to the right side:\n[\n9(x - 1)^2 - 16(y + 2)^2 = 144 - 55 = 89\n]", "The equation is now in the standard form of a hyperbola:\n[\n9(x - 1)^2 - 16(y + 2)^2 = 89\n]", "---", "### Solution and key insight", "This form reveals the conic’s center at $ (1, -2) $, since the squared terms are centered on these coordinates. Completing the square transformed the original equation into a clearer structure that identifies the vertical hyperbola’s center directly.", "---", "Final Answer:\n[\n\boxed{9(x - 1)^2 - 16(y + 2)^2 = 89}\n]\nThe center is at $ (1, -2) $", "---", "This method of completing the square is essential in conic section analysis, helping to reveal important geometric information hidden within algebra."]