["# Solving the Quadratic Equation: $$5x + 1 = x^2 + x - 2$$", "Understanding how to solve equations like $$5x + 1 = x^2 + x - 2$$ is essential in algebra and forms the foundation for more advanced mathematics, engineering, and physics problems. This article guides you step-by-step through solving this quadratic equation, explains how to identify its type, and explores real-world applications.", "---", "## Step 1: Rewrite Equation in Standard Quadratic Form", "The goal is to bring all terms to one side of the equation to form a standard quadratic expression:", "$$
\n5x + 1 = x^2 + x - 2
\n$$", "Subtract $ 5x + 1 $ from both sides:", "$$
\n0 = x^2 + x - 2 - (5x + 1)
\n$$", "Simplify the right-hand side:", "$$
\n0 = x^2 + x - 2 - 5x - 1
\n$$", "$$
\n0 = x^2 - 4x - 3
\n$$", "This is now in standard form:", "$$
\nx^2 - 4x - 3 = 0
\n$$", "---", "## Step 2: Classify the Equation", "This is a quadratic equation because the highest power of $ x $ is 2. Since the $ x^2 $ term has a positive coefficient, the parabola opens upwards, and the equation has two real roots (since the discriminant is positive).", "---", "## Step 3: Use the Quadratic Formula", "To solve $ x^2 - 4x - 3 = 0 $, use the quadratic formula:", "$$
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n$$", "Where $ a = 1 $, $ b = -4 $, and $ c = -3 $. Plug in the values:", "$$
\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-3)}}{2(1)}
\n$$", "$$
\nx = \frac{4 \pm \sqrt{16 + 12}}{2}
\n$$", "$$
\nx = \frac{4 \pm \sqrt{28}}{2}
\n$$", "$$
\n\sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}
\n$$", "So:", "$$
\nx = \frac{4 \pm 2\sqrt{7}}{2} = 2 \pm \sqrt{7}
\n$$", "---", "## Step 4: Final Solutions", "The two solutions to the equation $ 5x + 1 = x^2 + x - 2 $ are:", "$$
\nx = 2 + \sqrt{7} \quad \ ext{and} \quad x = 2 - \sqrt{7}
\n$$", "---", "## Why This Equation Matters", "Equations of the form $$ax^2 + bx + c = 0$$ appear frequently in:", "- Physics: modeling motion under acceleration
\n- Economics: profit maximization problems
\n- Engineering: designing curves and optimizing systems
\n- Computer Graphics: rendering conic sections", "Understanding how to solve them empowers you to tackle real-world problems involving parabolic relationships.", "---", "## Looking Ahead: Graphing the Equation", "The solutions $ 2 + \sqrt{7} \approx 4.65 $ and $ 2 - \sqrt{7} \approx -0.65 $ correspond to the $ x $-intercepts of the parabola $ y = x^2 - 4x - 3 $. These points help visualize the function’s behavior, including vertex location and symmetry.", "---", "## Key Takeaways", "- Always rewrite equations into standard quadratic form: $ ax^2 + bx + c = 0 $
\n- Use the quadratic formula when factoring is difficult
\n- The discriminant ($ b^2 - 4ac $) determines the nature of roots
\n- Real-world applications depend on modeling relationships with quadratic functions", "---", "Conclusion: Solving $ 5x + 1 = x^2 + x - 2 $ not only sharpens algebraic skills but connects to powerful tools in science and technology. With practice, quadratic equations become intuitive and accessible for both study and application."]
