\[ u = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2} \]

["Solving the Quadratic Equation: A Step-by-Step Guide to Finding ( u )", "Solving quadratic equations is a fundamental skill in algebra, essential for students, engineers, and anyone working with mathematical modeling. In this article, we’ll explore the quadratic formula applied to the equation:", "[\nu = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2}\n]", "This expression represents the two solutions of the quadratic equation written in standard form: ( au^2 + bu + c = 0 ). Let’s break down the equation, compute the discriminant, simplify, and find the precise values of ( u ).", "---", "### Step 1: Identify coefficients", "From the quadratic equation beneath the formula, identify:", "- ( a = 2 )\n- ( b = -3 )\n- ( c = -2 ) (Note: The equation is ( 2u^2 - 3u - 2 = 0 ))", "---", "### Step 2: Compute the discriminant", "The discriminant ( D ) dictates the nature of the roots and is calculated as:", "[\nD = b^2 - 4ac\n]", "Substitute the values:", "[\nD = (-3)^2 - 4 \cdot 2 \cdot (-2) = 9 + 16 = 25\n]", "Since ( D = 25 > 0 ), there are two distinct real solutions.", "---", "### Step 3: Apply the quadratic formula", "Plug ( a ), ( b ), and ( D ) into the quadratic formula:", "[\nu = \frac{-b \pm \sqrt{D}}{2a} = \frac{-(-3) \pm \sqrt{25}}{2 \cdot 2} = \frac{3 \pm 5}{4}\n]", "---", "### Step 4: Calculate the two solutions", "Compute both roots using ( \pm ):", "1.\n[\nu_1 = \frac{3 + 5}{4} = \frac{8}{4} = 2\n]", "2.\n[\nu_2 = \frac{3 - 5}{4} = \frac{-2}{4} = -\frac{1}{2}\n]", "---", "### Final Answer", "The solutions to the quadratic equation are:", "[\nu = 2 \quad \ ext{and} \quad u = -\frac{1}{2}\n]", "---", "### Why This Matters", "Mastering quadratic formulas enables students and professionals to solve real-world problems—from physics to finance—by modeling relationships that follow parabolic trends. Whether you’re optimizing profit, analyzing motion, or understanding physics motion equations, the ability to solve for ( u ) algebraically is invaluable.", "---", "### Summary", "Given the expression:", "[\nu = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2}\n]", "we calculated the discriminant, applied the quadratic formula, and found:", "[\n\boxed{u = 2 \quad \ ext{and} \quad u = -\frac{1}{2}}\n]", "Understanding how to derive and interpret these roots strengthens your algebraic foundation and prepares you for advanced problem-solving in mathematics and STEM fields.", "---", "Keywords: quadratic formula, solving quadratic equations, discriminant, squared root, algebra, step-by-step solution, real roots, ( u = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2} )", "---", "Explore further to enhance your math skills: practice solving different quadratics, review the meaning of the discriminant, and challenge yourself with word problems involving parabolic relationships."]