Expandiendo esto, \(x^2 - 3x + 2x - 6 = x^2 - x - 6 = 0\).

["Expanding and Solving the Quadratic Equation: (x^2 - 3x + 2x - 6 = 0)", "When solving quadratic equations, simplifying expressions before applying algebraic techniques can make equations easier to understand and solve. One such example is the equation:", "[\nx^2 - 3x + 2x - 6 = x^2 - x - 6 = 0\n]", "This equation may appear complex at first glance, but by expanding and simplifying it step by step, we uncover a straightforward quadratic equation ready for solution.", "---", "### Step 1: Simplify the Left-Hand Side", "Begin with the original expression:", "[\nx^2 - 3x + 2x - 6\n]", "Combine like terms:", "- Combine (-3x + 2x = -x)", "This simplifies the equation to:", "[\nx^2 - x - 6 = 0\n]", "---", "### Step 2: Factor the Quadratic", "Now, factor the simplified quadratic expression (x^2 - x - 6). We look for two numbers that multiply to (-6) (the constant term) and sum to (-1) (the coefficient of (x)).", "These numbers are (-3) and (+2):", "[\nx^2 - x - 6 = (x - 3)(x + 2)\n]", "---", "### Step 3: Solve the Factored Equation", "Set each factor equal to zero:", "[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]\n[\nx + 2 = 0 \quad \Rightarrow \quad x = -2\n]", "Thus, the solutions are (x = 3) and (x = -2).", "---", "### Why This Expansion Matters", "Simplifying expressions like (x^2 - 3x + 2x - 6) enhances clarity and reduces the risk of errors when solving. By combining like terms and factoring deliberately, you transform a seemingly complex equation into a standard quadratic form. This approach is especially valuable in algebra and higher-level math, where accurate simplification leads to correct and efficient solutions.", "---", "### Key Takeaways:", "- Combining like terms simplifies quadratic expressions.\n- Factoring quadratics efficiently relies on identifying correct factors.\n- Simplifying before solving improves accuracy and understanding.\n- (x^2 - x - 6 = 0) has solutions (x = 3) and (x = -2).", "---", "Expanding and systematically simplifying equations like (x^2 - 3x + 2x - 6 = 0) is a foundational skill in algebra, enabling deeper insight into quadratic behavior and equation solving techniques. Practice this expansion method to build stronger problem-solving abilities and prepare for more advanced mathematical challenges.", "---", "Keywords: expand quadratic equation, simplify (x^2 - 3x + 2x - 6), solve (x^2 - x - 6 = 0), factor quadratics, algebra tutorial, solve quadratic equations, step-by-step equation solving."]