\[ u = \frac{-3 \pm \sqrt{9 + 16}}{4} \]

["Titled: Solving the Quadratic Equation: Step-by-Step Guide to ( u = \frac{-3 \pm \sqrt{9 + 16}}{4} )", "Learning how to solve quadratic equations is essential for mastering algebra and building a strong foundation in mathematics. One frequently encountered expression involves computing solutions for a quadratic equation using the quadratic formula. In this article, we explore the solution to the equation expressed as:", "[\nu = \frac{-3 \pm \sqrt{9 + 16}}{4}\n]", "---", "### Understanding the Quadratic Formula", "The standard quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "Its solutions are given by:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "In our case, the expression ( u = \frac{-3 \pm \sqrt{9 + 16}}{4} ) arises directly from applying this formula. To recognize the components clearly:", "- ( a = 1 ) (implying ( b = -3 ), ( c = 16 ))\n- The discriminant is ( b^2 - 4ac = (-3)^2 - 4(1)(16) = 9 - 64 = -55 ), indicating two complex roots.\n- The denominator ( 2a = 4 ), matching the given expression.", "---", "### Step-by-Step Solution", "Start with the quadratic formula:", "[\nu = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(16)}}{4}\n]", "Simplify the components:", "- ( -(-3) = +3 )\n- Discriminant: ( 9 - 64 = -55 )", "So,", "[\nu = \frac{3 \pm \sqrt{-55}}{4}\n]", "Since the discriminant is negative, the solutions involve imaginary numbers. Rewrite ( \sqrt{-55} ) as ( i\sqrt{55} ), where ( i = \sqrt{-1} ).", "Thus,", "[\nu = \frac{3 \pm i\sqrt{55}}{4}\n]", "---", "### Final Answer", "The two complex solutions to the equation are:", "[\nu = \frac{3}{4} + \frac{\sqrt{55}}{4}i \quad \ ext{and} \quad u = \frac{3}{4} - \frac{\sqrt{55}}{4}i\n]", "---", "### Why This Equation Matters", "This solution demonstrates how the quadratic formula applies even when the discriminant is negative — the key lies in understanding complex numbers. Applying this concept is vital in physics, engineering, and computer science, where modeling real-world phenomena often requires handling complex results.", "---", "### Tips for Solving Similar Equations", "1. Identify coefficients ( a ), ( b ), and ( c ) carefully from the quadratic form.\n2. Compute ( b^2 - 4ac ) accurately to determine the nature of roots (real or complex).\n3. Simplify radicals and rational expressions fully before writing the final answer.\n4. Express solutions clearly in standard form: ( u = \frac{\ ext{real part} \pm i,\ ext{imaginary part}}{4} ).", "---", "### Conclusion", "The expression ( u = \frac{-3 \pm \sqrt{9 + 16}}{4} ) elegantly illustrates the power of the quadratic formula. With a little algebraic manipulation, you’ll find complex conjugate roots that showcase the beauty and completeness of algebra in both real and imaginary domains.", "---", "Keywords: quadratic equation, quadratic formula, complex numbers, discriminant, ( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), imaginary solutions, algebra tutorial, solving quadratics.\nMeta Description: Solve ( u = \frac{-3 \pm \sqrt{9 + 16}}{4} ) using the quadratic formula. Learn step-by-step how to find complex roots and understand their mathematical significance. Ideal for students and educators."]