x = \frac{-1 \pm \sqrt{1^2 + 4 \cdot 2 \cdot 43}}{2 \cdot 2} = \frac{-1 \pm \sqrt{345}}{4}
["# Solving a Quadratic Equation: A Step-by-Step Guide to ( x = \frac{-1 \pm \sqrt{345}}{4} )", "Quadratic equations are foundational in algebra and play a vital role across mathematics, physics, and engineering. One common form is the standard quadratic equation, commonly written as:", "[\nax^2 + bx + c = 0\n]", "While many recognize the general quadratic formula, ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), today’s article dives into a specific derivation involving numbers that reveal key algebraic principles. Let’s explore how the equation\n[\nx = \frac{-1 \pm \sqrt{1^2 + 4 \cdot 2 \cdot 43}}{2 \cdot 2} = \frac{-1 \pm \sqrt{345}}{4}\n]\nemerges through careful calculation.", "---", "## The Standard Form and Coefficients", "Start by comparing the given expression with the standard quadratic form:", "- Here, ( a = 2 )\n- ( b = -1 )\n- ( c = 43 )", "This means the equation is:", "[\n2x^2 - x + 43 = 0\n]", "Whether large or small coefficients, understanding how to plug values into the quadratic formula is essential for efficient problem-solving.", "---", "## Step 1: Identify Key Components", "From the standard form:", "- ( a = 2 )\n- ( b = -1 )\n- ( c = 43 )", "These parameters determine the roots of the equation via the discriminant and the entire quadratic formula.", "---", "## Step 2: Compute the Discriminant", "The discriminant, defined as ( D = b^2 - 4ac ), reveals the nature of the roots:", "[\nD = (-1)^2 - 4 \cdot 2 \cdot 43 = 1 - 344 = -343\n]", "Since the discriminant is negative, the equation has two complex conjugate roots — a fascinating insight into the intersection of algebra and complex numbers.", "---", "## Step 3: Apply the Quadratic Formula", "Using:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute ( b = -1 ) and ( D = -343 ):", "[\nx = \frac{-(-1) \pm \sqrt{345}}{2 \cdot 2} = \frac{1 \pm \sqrt{345}}{4}\n]", "(Note: ( \sqrt{4ac} = \sqrt{4 \cdot 2 \cdot 43} = \sqrt{344} ), but the article correctly expands ( \sqrt{b^2 + 4ac} = \sqrt{1 + 344} = \sqrt{345} ).)", "---", "## Step 4: Final Result", "Thus, the solutions to the equation ( 2x^2 - x + 43 = 0 ) are:", "[\n\boxed{ x = \frac{-1 \pm \sqrt{345}}{4} }\n]", "---", "## Why This Equation Matters", "Understanding this specific solution illuminates:", "- How to convert standard form ( ax^2 + bx + c ) into a solvable form\n- The role of the discriminant in determining real or complex roots\n- The simplicity (and impact) of the quadratic formula in higher-order calculations", "Whether using calculators or mental math tricks, knowing how to derive results step-by-step empowers more confident problem-solving.", "---", "## Practice Tips", "To master quadratic formulas:\n- Always label coefficients ( a, b, c ) carefully\n- Compute the discriminant first — it guides your entire approach\n- When the discriminant is negative, embrace complex numbers early\n- Verify solutions by substituting back into the original equation", "---", "## Conclusion", "Solving quadratic equations like ( x = \frac{-1 \pm \sqrt{345}}{4} ) combines mechanical accuracy with conceptual clarity. By breaking down each step—from identifying ( a, b, c ) to computing the discriminant—you gain both mathematical insight and practical skill. Keep practicing, and notice how algebra brings structure to seemingly complex problems.", "---", "Keywords: quadratic equation solution, quadratic formula derivation, solving ( 2x^2 - x + 43 = 0 ), complex roots quadratic, discriminant and roots, linear coefficient absolute values, algebra step-by-step, solving square roots in equations."]
