x = [-21 ± √(441 + 288)] / 4 - United Radiology

["### Solving the Quadratic Equation: x = [-21 ± √(441 + 288)] / 4", "Solving quadratic equations is a fundamental skill in algebra, with wide-ranging applications in physics, engineering, economics, and computer science. One particularly instructive example is the equation:", "[
\nx = \frac{-21 \pm \sqrt{441 + 288}}{4}
\n]", "This equation presents a classic quadratic form and demonstrates how the quadratic formula helps find precise solutions, even when inside the square root includes a positive additive expression.", "---", "### Step-by-Step Breakdown of the Equation", "The general quadratic formula solves equations of the form:", "[
\nax^2 + bx + c = 0
\n]", "The solutions are:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Comparing with your equation:", "- Here, ( a = 1 ) (implied, since no ( x^2 ) coefficient appears explicitly),
\n- ( b = -21 ),
\n- ( c = 288 ).", "However, in the expression ( x = \frac{-21 \pm \sqrt{441 + 288}}{4} ), the radicand is ( 441 + 288 ). This suggests ( b^2 + 4ac = 441 + 288 = 729 ), meaning:", "[
\nb^2 - 4ac = 729
\n]", "So, the equation behaves effectively like:", "[
\nx = \frac{-b \pm \sqrt{b^2 + 4ac}}{2a} \quad \ ext{with } b = -21,\ a = 1,\ c = 288
\n]", "This arises often when completing the square or when the discriminant is shaped as a sum rather than a difference.", "---", "### Simplify and Compute the Discriminant", "Compute the value under the square root:", "[
\n441 + 288 = 729
\n]", "And ( \sqrt{729} = 27 ), a perfect square that makes this solvable without decimal approximations.", "---", "### Apply the Quadratic Formula", "Substitute values into the formula:", "[
\nx = \frac{-(-21) \pm \sqrt{729}}{4} = \frac{21 \pm 27}{4}
\n]", "This gives two solutions:", "- First root:
\n [
\n x_1 = \frac{21 + 27}{4} = \frac{48}{4} = 12
\n ]", "- Second root:
\n [
\n x_2 = \frac{21 - 27}{4} = \frac{-6}{4} = -\frac{3}{2}
\n ]", "---", "### Summary of Solutions", "The equation
\n[
\nx = \frac{-21 \pm \sqrt{441 + 288}}{4}
\n]
\nhas two real solutions:
\n[
\nx = 12 \quad \ ext{and} \quad x = -\frac{3}{2}
\n]", "These solutions reflect how quadratic equations can yield both rational and irrational (when discriminant is not perfect) results. In this case, the positive, clean discriminant simplifies neatly.", "---", "### Why This Equation Matters", "1. Algebraic Clarity: Demonstrates form and structure of quadratic solutions with non-standard radicands.
\n2. Practical Use: Solving équations with sums under square roots often appears in displacement problems, optimization models, and graphing applications.
\n3. Perfect Square Pattern: The discriminant ( 441 + 288 = 729 = 27^2 ) showcases how recognizing perfect squares accelerates simplification.", "---", "### Concluding Thoughts", "Mastering quadratics like ( x = \frac{-21 \pm \sqrt{441 + 288}}{4} ) sharpens algebraic intuition and prepares learners to tackle complex equations confidently. Whether you’re balancing budgets, modeling motion, or analyzing data trends, the quadratic formula remains an indispensable tool.", "---", "Keywords: quadratic equation solution, solving quadratics, x = [-21 ± √(441 + 288)] / 4, algebraic steps, discriminant calculation, real roots, perfect square discriminant, mathematical problem-solving."]