\(x = \frac{3 \pm \sqrt{(-3)^2 - 4 \times 2 \times (-2)}}{4}\).
["# Solving the Quadratic Equation: Understanding (x = \dfrac{3 \pm \sqrt{(-3)^2 - 4 \ imes 2 \ imes (-2)}}{4})", "Solving quadratic equations is a fundamental skill in algebra, essential for students, engineers, and data scientists alike. One excellent example that demonstrates the quadratic formula in action is the equation:", "[\nx = \dfrac{3 \pm \sqrt{(-3)^2 - 4 \ imes 2 \ imes (-2)}}{4}\n]", "Understanding how to apply this formula unlocks deeper insights into the nature of quadratic functions, their roots, and their applications in real-world modeling.", "## Step-by-Step Breakdown of the Equation", "The general form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "In our expression, by identifying coefficients (a), (b), and (c):", "- (a = 2)\n- (b = 3)\n- (c = -2)", "The quadratic formula is:", "[\nx = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Now substitute the values into the formula:", "[\nx = \dfrac{ -3 \pm \sqrt{(3)^2 - 4 \ imes 2 \ imes (-2)} }{4}\n]", "Simplify the discriminant ((-3)^2 - 4 \ imes 2 \ imes (-2)):", "- ( (-3)^2 = 9 )\n- ( 4 \ imes 2 \ imes (-2) = -16 ), so minus this is ( +16 )", "Thus, the discriminant is:", "[\n9 + 16 = 25\n]", "Now substitute back:", "[\nx = \dfrac{-3 \pm \sqrt{25}}{4}\n]", "Since (\sqrt{25} = 5), this simplifies further:", "[\nx = \dfrac{-3 \pm 5}{4}\n]", "This gives two solutions:", "- (x = \dfrac{-3 + 5}{4} = \dfrac{2}{4} = \dfrac{1}{2})\n- (x = \dfrac{-3 - 5}{4} = \dfrac{-8}{4} = -2)", "So the solutions are:", "[\nx = \dfrac{1}{2} \quad \ ext{and} \quad x = -2\n]", "## Why Is This Equation Important?", "This quadratic equation exemplifies key concepts:", "- Real and distinct roots due to a positive discriminant ((25 > 0))\n- Rational solutions since the discriminant is a perfect square\n- The impact of the numerator ( \pm \sqrt{D} ) in generating two solutions\n- Use in modeling scenarios where two possible outcomes depend on a variable parameter", "## Practical Applications of Quadratic Solutions", "Quadratic equations appear in physics (projectile motion), economics (revenue models), engineering (structural calculations), and computer graphics (parabolic curves). Understanding how to solve equations like this empowers you to model and analyze systems involving quadratic behavior.", "## Conclusion", "The expression (x = \dfrac{3 \pm \sqrt{(-3)^2 - 4 \ imes 2 \ imes (-2)}}{4}) is not just a computation — it’s a gateway to mastering quadratic solutions. By systematically applying the quadratic formula, expanding the discriminant, and simplifying step by step, you uncover the precise values that define the equation’s behavior. Whether you’re a student learning algebra or a professional solving applied math problems, mastering this process builds confidence and clarity in math.", "For further practice, try similar problems adjusting coefficients or evaluating roots numerically—this strengthens algebraic intuition and prepares you for advanced calculus and applied mathematics.", "---", "Keywords: quadratic formula, solving quadratics, discriminant, (x = \dfrac{3 \pm \sqrt{45}}{4}), algebra tutorial, quadratic equations, mathematical solutions, rational roots, real roots quadratic, step-by-step algebra."]
