A quadratic equation \(2x^2 - 3x - 2 = 0\) is given. Find its roots.
["SEO-Optimized Article: Finding the Roots of the Quadratic Equation (2x^2 - 3x - 2 = 0)", "Solving quadratic equations is a fundamental skill in algebra, essential for students, engineers, and math enthusiasts. In this article, we’ll explore how to find the roots of the quadratic equation:", "[\n2x^2 - 3x - 2 = 0\n]", "Understanding the roots helps in graphing parabolas, modeling real-world problems, and solving complex equations. This guide walks you step-by-step through finding the roots using the quadratic formula—maximizing clarity and searchability for students seeking accurate solutions.", "---", "### What is a Quadratic Equation?", "A quadratic equation is any equation of the form:", "[\nax^2 + bx + c = 0\n]", "where (a), (b), and (c) are constants and (a <br/>\neq 0). The roots (or solutions) of this equation represent the (x)-values where the parabola intersects the (x)-axis.", "For our example,\n(a = 2), (b = -3), (c = -2).", "---", "### Step-by-Step: Finding the Roots Using the Quadratic Formula", "The most reliable method for solving quadratic equations is the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Let’s compute each part for (2x^2 - 3x - 2 = 0).", "#### 1. Identify coefficients\nFrom (2x^2 - 3x - 2):\n- (a = 2)\n- (b = -3)\n- (c = -2)", "#### 2. Compute the discriminant ((D))\nThe discriminant determines the nature of the roots:\n[\nD = b^2 - 4ac\n]", "Substitute the values:", "[\nD = (-3)^2 - 4(2)(-2) = 9 + 16 = 25\n]", "Since (D = 25 > 0), there are two distinct real roots.", "#### 3. Plug values into the quadratic formula", "[\nx = \frac{-(-3) \pm \sqrt{25}}{2(2)} = \frac{3 \pm 5}{4}\n]", "Now calculate both solutions:", "- (x_1 = \frac{3 + 5}{4} = \frac{8}{4} = 2)\n- (x_2 = \frac{3 - 5}{4} = \frac{-2}{4} = -\frac{1}{2})", "---", "### Final Result: The Roots of (2x^2 - 3x - 2 = 0)", "The solutions are:", "[\nx = 2 \quad \ ext{and} \quad x = -\frac{1}{2}\n]", "---", "### Why This Equation Matters", "Knowing the roots provides key insights:\n- The parabola (y = 2x^2 - 3x - 2) crosses the (x)-axis at (x = 2) and (x = -0.5).\n- It opens upward because (a = 2 > 0).\n- This form is useful in physics, economics, and engineering for modeling relationships and optimizing outcomes.", "---", "### Summary", "To solve any quadratic equation:\n1. Identify coefficients (a), (b), and (c)\n2. Compute the discriminant (D = b^2 - 4ac)\n3. Use the quadratic formula:\n [\n x = \frac{-b \pm \sqrt{D}}{2a}\n ]\n4. Simplify to find the exact roots", "This method works for all quadratic equations and is the cornerstone of solving second-degree polynomial problems—critical for mastering algebra (high school level and beyond).", "---", "Keywords for SEO:\nquadratic equation roots, solve (2x^2 - 3x - 2 = 0), quadratic formula step-by-step, find roots of a quadratic, discriminant meaning, real solutions of quadratics, algebra tutorial", "Meta Description:\nLearn how to find the roots of (2x^2 - 3x - 2 = 0) using the quadratic formula. Step-by-step solution with explanation, discriminant analysis, and real-world relevance. Perfect for students mastering algebra.", "---", "Takeaway:\nMastering how to solve (2x^2 - 3x - 2 = 0) builds a strong foundation for advanced math and engineering applications. Use the quadratic formula confidently—steps are clear, outcomes are precise, and understanding is powerful.", "---", "Did you solve this equation before? Try it using both the quadratic formula and factoring methods—comparison boosts mastery! Reflect, verify, and practice deeply. Your algebra journey starts here."]
