First equation: $ -x^2 + y^2 + 2x - 4y = 13 $.

["# First Equation: $ -x^2 + y^2 + 2x - 4y = 13 $ — Step-by-Step Solution and Analysis", "Understanding and solving quadratic equations in two variables can seem challenging at first, but with the right approach, you can simplify and interpret them effectively. This article explores the first equation:\n$$\n -x^2 + y^2 + 2x - 4y = 13\n$$\nfrom a clear, methodical perspective—ideal for anyone learning about conic sections, quadratic transformations, or algebraic manipulation.", "---", "## What Is This Equation?", "The given equation\n$$\n -x^2 + y^2 + 2x - 4y = 13\n$$\nis a second-degree equation in two variables $x$ and $y$, featuring both $x^2$ and $y^2$ with unlike signs. This structure indicates a hyperbola, but we’ll guide you through identifying and solving it step by step.", "---", "## Step 1: Rewrite the Equation in Standard Form", "Start by rearranging all terms to one side, organizing terms with coefficients:", "$$\ny^2 - x^2 + 2x - 4y = 13\n$$", "We recognize $y^2 - x^2$ as a difference of squares, suggesting a hyperbolic form. However, free linear $x$ and $y$ terms (2x and -4y) mean we must complete the square to eliminate linear terms and rewrite the equation in a clearer canonical form.", "---", "## Step 2: Complete the Square", "Group $y$-terms and $x$-terms:", "$$\n(y^2 - 4y) + (-x^2 + 2x) = 13\n$$", "Complete the square for $y$:\n$y^2 - 4y = (y - 2)^2 - 4$", "Complete the square for $x$:\n$-x^2 + 2x = - (x^2 - 2x) = -[(x - 1)^2 - 1] = - (x - 1)^2 + 1$", "Substitute back:", "$$\n(y - 2)^2 - 4 - (x - 1)^2 + 1 = 13\n$$", "Simplify constants:", "$$\n(y - 2)^2 - (x - 1)^2 - 3 = 13\n$$", "$$\n(y - 2)^2 - (x - 1)^2 = 16\n$$", "---", "## Step 3: Identify the Conic Section", "Now the equation is:", "$$\n(y - 2)^2 - (x - 1)^2 = 16\n$$", "This matches the standard form of a hyperbola centered at $(h, k) = (1, 2)$, opening vertically (since $y^2$ has a positive coefficient):", "$$\n\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\n$$", "Rewriting ours:", "$$\n\frac{(y - 2)^2}{16} - \frac{(x - 1)^2}{16} = 1\n$$", "So $a^2 = 16 \Rightarrow a = 4$, $b^2 = 16 \Rightarrow b = 4$. The hyperbola opens up and down along the $y$-axis.", "---", "## Step 4: Analyze Key Features", "- Center: $(1, 2)$\n- Transverse axis: Vertical\n- Conjugate axis: Horizontal\n- Vertices: $(1, 2 \pm a) = (1, 6)$ and $(1, -2)$\n- Asymptotes: $y - 2 = \pm \frac{a}{b}(x - 1) = \pm(x - 1)$", "That is, the asymptotes are the lines:\n$$\ny = x + 1 \quad \ ext{and} \quad y = -x + 1\n$$", "---", "## Step 5: Solve for $y$ or $x$ (Optional)", "You can solve for one variable:", "From $(y - 2)^2 - (x - 1)^2 = 16$, isolate $y$:", "$$\n(y - 2)^2 = (x - 1)^2 + 16\n$$", "$$\ny - 2 = \pm \sqrt{(x - 1)^2 + 16}\n$$", "$$\ny = 2 \pm \sqrt{(x - 1)^2 + 16}\n$$", "This form shows two branches opening vertically—as expected for a vertical hyperbola.", "---", "## Why This Equation Matters", "Understanding equations like $ -x^2 + y^2 + 2x - 4y = 13 $ helps you:", "- Recognize hyperbolas and interpret their orientation and center\n- Apply algebraic techniques to transform and simplify conic sections\n- Model real-world phenomena such as orbital paths, signal propagation, or engineering tolerances", "---", "## Conclusion", "The equation\n$$\n -x^2 + y^2 + 2x - 4y = 13\n$$\nrepresents a vertical hyperbola centered at $(1, 2)$, derived through completing the square and identifying standard form. By mastering this process, you gain insight into conic geometry and expand your toolkit for analyzing quadratic relationships in two dimensions.", "Whether for homework, exams, or self-study, this method—completing the square, recognizing conic forms, and applying algebraic transformations—is essential knowledge in algebra and precalculus.", "---", "### Keywords for SEO:\n- Horizontal and vertical hyperbola equation\n- Completing the square for conic sections\n- First equation $ -x^2 + y^2 + 2x - 4y = 13 $\n- How to solve quadratic equations in two variables\n- Identify hyperbola from $y^2 - x^2 + \dots$\n- Standard form of hyperbola\n- Conic section practice problems", "---", "### Learn More:\n- Explore asymptotes and foci of hyperbolas\n- Compare horizontal vs vertical hyperbolas\n- Practice converting general conic equations to standard form", "---", "Tagline: Master the first equation — $ -x^2 + y^2 + 2x - 4y = 13 $ — with precision and clarity. Understand its structure, solve it step-by-step, and unlock the world of conic sections!"]