A science administrator is reviewing a grant application involving the hyperbola given by the equation \( 9x^2 - 16y^2 - 54x + 64y - 71 = 0 \). Find the center of this hyperbola.

["Understanding the Hyperbola: Finding the Center Using Completing the Square", "When studying conic sections in advanced mathematics and research, identifying the geometric center of a hyperbola is essential for modeling, data analysis, and scientific simulations. A recent grant application examined a hyperbola defined by the equation:\n[\n9x^2 - 16y^2 - 54x + 64y - 71 = 0\n]\nThis article explains how a science administrator reviewing such a grant application can solve for the hyperbola’s center using analytical geometry.", "### Step 1: Rewrite the Equation in Standard Form\nTo determine the center, we must rewrite the equation into the standard form of a hyperbola by completing the square. Start with the original equation:\n[\n9x^2 - 16y^2 - 54x + 64y - 71 = 0\n]", "Group the (x)-terms and (y)-terms:\n[\n(9x^2 - 54x) - (16y^2 - 64y) = 71\n]", "Factor out coefficients of squared terms:\n[\n9(x^2 - 6x) - 16(y^2 - 4y) = 71\n]", "### Step 2: Complete the Square\nComplete the square inside each group.", "For (x):\n[\nx^2 - 6x \quad \Rightarrow \quad (x - 3)^2 - 9\n]\nSo,\n[\n9(x^2 - 6x) = 9\left[(x - 3)^2 - 9\right] = 9(x - 3)^2 - 81\n]", "For (y):\n[\ny^2 - 4y \quad \Rightarrow \quad (y - 2)^2 - 4\n]\nSo,\n[\n-16(y^2 - 4y) = -16\left[(y - 2)^2 - 4\right] = -16(y - 2)^2 + 64\n]", "Substitute back into the equation:\n[\n9(x - 3)^2 - 81 - 16(y - 2)^2 + 64 = 71\n]", "Simplify constants:\n[\n9(x - 3)^2 - 16(y - 2)^2 - 17 = 71\n]\n[\n9(x - 3)^2 - 16(y - 2)^2 = 88\n]", "### Step 3: Divide to Obtain Standard Form\nDivide both sides by 88:\n[\n\frac{9(x - 3)^2}{88} - \frac{16(y - 2)^2}{88} = 1\n\quad \Rightarrow \quad\n\frac{(x - 3)^2}{\frac{88}{9}} - \frac{(y - 2)^2}{\frac{88}{16}} = 1\n]", "This is now in the standard form of a hyperbola:\n[\n\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\n]\nwhere ((h, k)) is the center.", "### Step 4: Identify the Center\nFrom the completed square form, the center is clearly:\n[\n(h, k) = (3, 2)\n]", "### Conclusion\nFor the grant application under review, the science administrator can confidently confirm that the center of the hyperbola defined by ( 9x^2 - 16y^2 - 54x + 64y - 71 = 0 ) is at ((3, 2)). This precise geometric insight supports accurate modeling in fields ranging from physics to data science, underscoring the importance of rigorous analysis in scientific research.", "Keywords: hyperbola center, complete the square, conic sections, hyperbola equation, (9x^2 - 16y^2 - 54x + 64y - 71 = 0), science administrator review, coordinate geometry, analytical geometry."]