y = \frac{-42 \pm \sqrt{1764 - 1512}}{54} = \frac{-42 \pm \sqrt{252}}{54} = \frac{-42 \pm 6\sqrt{7}}{54} = \frac{-7 \pm \sqrt{7}}{9}.

["# Solving Quadratic Equations: A Deep Dive into y = (−42 ± √(1764 − 1512)) / 54", "Understanding how to solve quadratic equations is fundamental in algebra, serving as a building block for advanced mathematics and various real-world applications. One common problem you may encounter involves expressions like:", "[\ny = \frac{-42 \pm \sqrt{1764 - 1512}}{54}\n]", "In this article, we’ll walk step-by-step through simplifying this expression, solving the corresponding quadratic equation, and fully interpreting the solution set.", "---", "## Step 1: Simplify the Discriminant Inside the Square Root", "The discriminant of a quadratic equation in the form ( ax^2 + bx + c = 0 ) is given by ( D = b^2 - 4ac ). In our case:", "- ( a = 1 ), ( b = -42 ), ( c = -1512 ) (though here coefficients derive from the discriminant directly)\n- So, compute:\n[\nD = (-42)^2 - 4(1)(-1512) = 1764 + 6048 = 7812\n]", "However, the expression given simplifies the square root term as:", "[\n\sqrt{1764 - 1512} = \sqrt{252}\n]", "Indeed:\n[\n1764 - 1512 = 252\n]", "Further simplifying ( \sqrt{252} ):", "Factor ( 252 = 36 \ imes 7 ), so\n[\n\sqrt{252} = \sqrt{36 \cdot 7} = 6\sqrt{7}\n]", "Hence, the original expression simplifies beautifully to:\n[\ny = \frac{-42 \pm 6\sqrt{7}}{54}\n]", "---", "## Step 2: Simplify the Fraction", "We can simplify the entire expression by dividing numerator and denominator by the greatest common divisor (GCD) of ( -42 \pm 6\sqrt{7} ) and 54.", "Note that all terms are divisible by 6:", "[\ny = \frac{6(-7 \pm \sqrt{7})}{6 \cdot 9} = \frac{-7 \pm \sqrt{7}}{9}\n]", "Perfect! The simplified solution is:\n[\ny = \frac{-7 \pm \sqrt{7}}{9}\n]", "---", "## Step 3: Why This Simplification Matters", "While the form\n[\ny = \frac{-42 \pm 6\sqrt{7}}{54}\n]\nis algebraically valid, simplifying it to\n[\ny = \frac{-7 \pm \sqrt{7}}{9}\n]\nis preferred for several reasons:", "- Clarity: The simplified expression clearly reveals the structure tied to the quadratic formula.\n- Ease of Use: Smaller numbers reduce the chance of arithmetic errors in calculations.\n- Eigenvalue & Function Analysis: This form makes it easier to analyze roots, graphs, and domain behavior.", "---", "## Step 4: Solving the Corresponding Quadratic Equation", "To understand why this simplified expression arises, consider the original quadratic equation:", "[\nx^2 + 42x - 1512 = 0\n]", "Using the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-42 \pm \sqrt{1764 + 6048}}{2 \cdot 1} = \frac{-42 \pm \sqrt{7812}}{2}\n]", "Earlier, we found the discriminant simplifies to 252, not 7812 — wait! Actually, ( \sqrt{1764 + 6048} = \sqrt{7812} ), but the problem states ( \sqrt{1764 - 1512} ), which was computed as ( \sqrt{252} ). This suggests a critical observation:", "Wait — correction:\n[\n1764 - 1512 = 252 \Rightarrow D = 252\n]\nBut ( \sqrt{252} = 6\sqrt{7} ), so:", "The correct simplified form is:\n[\nx = \frac{-42 \pm \sqrt{252}}{2} = \frac{-42 \pm 6\sqrt{7}}{2} = -21 \pm 3\sqrt{7}\n]", "But this contradicts the given simplified form ( \frac{-7 \pm \sqrt{7}}{9} ).\n👉 Important Insight: The expression in the problem is not derived from the equation ( x^2 + 42x - 1512 = 0 ). Instead, it appears to be a simplified result — perhaps from a normalized or rescaled problem.", "---", "## Step 5: Real-World Interpretation & Use Cases", "Simplified radical forms like ( y = \frac{-7 \pm \sqrt{7}}{9} ) appear frequently in:", "- Physics: When solving for motion equations involving square roots\n- Engineering: In signal processing or control theory\n- Economics: Modeling nonlinear cost or revenue functions\n- Geometry: Finding distances involving irrational coordinates", "The presence of ( \sqrt{7} ) often signals geometric origins (e.g., 45°–60°–75° triangles) or algebraic completeness in irrational solution sets.", "---", "## Step 6: Final Thoughts", "Mastering quadratic expressions goes beyond mere computation — it’s about understanding how simplifications reveal deeper structure. While multiple forms may describe the same solution set, reducing ( \frac{-42 \pm \sqrt{1764 - 1512}}{54} ) to ( \frac{-7 \pm \sqrt{7}}{9} ) enhances clarity and accessibility.", "Whether you’re solving equations by hand, graphing functions, or applying formulas in higher math, recognizing these simplifications equips you with precision and confidence.", "---", "Summary\n- The expression simplifies from ( \frac{-42 \pm \sqrt{252}}{54} ) to ( \frac{-7 \pm \sqrt{7}}{9} ) via factoring.\n- The discriminant ( D = 252 = 36 \ imes 7 ) enables elegant radical simplification.\n- Simplified forms improve readability and facilitate further mathematical analysis.\n- Real-world applications highlight the importance of mastering such transformations.", "---", "Keywords: quadratic formula, simplify radicals, solve quadratic equations, discriminant simplification, ( y = \frac{-7 \pm \sqrt{7}}{9} ), rationalizing expressions, algebraic simplification.", "For more insights into quadratic equations and their elegant solutions, explore step-by-step tutorials and practice problems on algebraic techniques."]