x = [-2 ± √(4 + 560)]/2 = [-2 ± √564]/2 - United Radiology

["# Solving Quadratic Equations: Understanding the Expression
\nx = [-2 ± √(4 + 560)]/2 = [-2 ± √564]/2", "Quadratic equations form a fundamental part of algebra and have wide applications in physics, engineering, and economics. One common form involves expressions derived from the quadratic formula:
\nx = [-b ± √(b² - 4ac)] / (2a)
\nIn this article, we’ll break down a specific quadratic expression—x = [-2 ± √(4 + 560)]/2—and walk you through its step-by-step solution, along with an explanation of key algebraic concepts.", "---", "## Understanding the Structure: The Quadratic Formula", "At the heart of solving quadratic equations lies the standard form:
\nax² + bx + c = 0
\nUsing this, the quadratic formula gives the solutions:
\nx = [-b ± √(b² - 4ac)] / (2a)", "In the given expression:
\nx = [-2 ± √(4 + 560)]/2, we can identify the coefficients by matching:
\n- ( b = 2 )
\n- ( 4ac = b² + 560 = 2² + 560 = 4 + 560 = 564 )
\nThus, a = 1, since ( a ) is the coefficient of ( x^2 ) and here ( a = 1 ).", "---", "## Step-by-Step Calculation", "### Step 1: Identify coefficients
\nFrom x = [-2 ± √(4 + 560)]/2
\nWe rewrite under the radical as:
\n[
\nx = \dfrac{ -2 ± √(4 + 560) }{2}
\n]", "### Step 2: Simplify under the square root
\nCalculate the constant under the radical:
\n[
\n4 + 560 = 564
\n]
\nSo the expression becomes:
\n[
\nx = \dfrac{ -2 ± √564 }{2}
\n]", "### Step 3: Simplify the expression
\nFactor numerator and denominator if possible. Here, factor √564:
\nCheck for perfect square factors of 564.", "Factor 564:
\n[
\n564 = 4 × 141 = 2² × 141
\n]
\nThus:
\n[
\n√564 = √(2² × 141) = 2√141
\n]", "Substitute back:
\n[
\nx = \dfrac{ -2 ± 2√141 }{2}
\n]", "### Step 4: Final simplification
\nDivide numerator terms by 2:
\n[
\nx = -1 ± √141
\n]", "---", "## Final Answer", "[
\n\boxed{ x = -1 \pm √141 }
\n]", "This matches the expected form, showing two solutions:
\n[
\nx = -1 + √141 \quad \ ext{and} \quad x = -1 - √141
\n]", "---", "## Why This Expression Matters", "This simplified form demonstrates key algebraic principles:
\n- Completing the discriminant simplification.
\n- Simplifying square roots for cleaner, more interpretable solutions.
\n- How numerical manipulations reduce complex roots into recognizable forms.", "Understanding expressions like x = [-2 ± √564]/2 supports logical reasoning in more advanced math, from calculus to engineering problem-solving.", "---", "## Key Takeaways", "- Always identify a, b, c carefully to apply the quadratic formula correctly.
\n- Simplifying radicals improves readability and opens paths for exact solutions.
\n- Expressing answers in ± form captures both possible real roots.
\n- Algebraic manipulation turns complex square roots into manageable expressions.", "Whether you're solving equations for homework, studying for exams, or building foundational math skills, mastering this type of expression strengthens your analytical toolkit.", "---", "Keywords: quadratic formula, solving x with radicals, simplifying √564, exact solutions, algebraic manipulation, discriminant, x = [-2 ± √(4 + 560)]/2, x = -1 ± √141, math tutoring, algebra problems, quantum math learning", "---", "For more insights into quadratic equations and related topics, keep exploring our resources on Khan Academy-style algebra tutorials and problem-solving tips."]