Using quadratic formula: x = [-2 ± √(4 + 44)] / 2 - United Radiology

["Using the Quadratic Formula: Solving x = [−2 ± √(4 + 44)] / 2", "The quadratic formula is one of the most powerful tools in algebra, enabling students and professionals alike to solve quadratic equations of the form ax² + bx + c = 0 efficiently. In this article, we’ll explore how to apply the quadratic formula step-by-step using the example equation:", "$$ x = \frac{-2 \pm \sqrt{4 + 44}}{2} $$", "---", "### Understanding the Quadratic Formula", "The general quadratic equation is:", "$$ ax^2 + bx + c = 0 $$", "The quadratic formula gives the solutions for x:", "$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$", "In our example, rewrite the expression under the square root in the form of $ b^2 - 4ac $:", "Given:
\n$$ x = \frac{-2 \pm \sqrt{4 + 44}}{2} $$", "Observe that $ -2 $ corresponds to –b when b = 2, and 4 + 44 = 48 represents the discriminant $ b^2 - 4ac $, so:", "$$
\nb^2 - 4ac = 4 + 44 = 48
\n$$", "---", "### Step-by-Step Solution", "1. Identify coefficients
\n From the quadratic expression, we determine:
\n - $ a = 1 $
\n - $ b = 2 $
\n - $ c = ? $ — since the equation is $ x = \frac{-2 \pm \sqrt{4 + 44}}{2} $, comparing with $ b^2 - 4ac $, and knowing $ b^2 = 4 $, we see that:
\n $$ b^2 - 4ac = 4 + 44 = 48 \Rightarrow 4 - 4ac = 48? \quad \ ext{Wait — this indicates a correction needed.} $$", "Actually, the term under the root is $ 4 + 44 = 48 $, so comparing:
\n $ b^2 - 4ac = 48 $, and since $ b = 2 $, $ b^2 = 4 $, then:
\n $$
\n 4 - 4ac = 48 \Rightarrow -4ac = 44 \Rightarrow ac = -11
\n $$
\n But this suggests inconsistency unless we revisit the setup.", "Let’s double-check: the expression given is $ \sqrt{4 + 44} = \sqrt{48} $, and from standard quadratic form $ ax^2 + x + c $, we usually have $ b = 1 $, but here $ b = 2 $, so mismatch.", "Clarification: The equation must be interpreted carefully. Since the formula shows $ x = \frac{-2 \pm \sqrt{4 + 44}}{2} $, this suggests the original equation may have been derived as:", "$$
\n x^2 + 2x + c = 0 \quad \ ext{with discriminant} \quad (2)^2 - 4(1)(c) = 4 - 4c = 48
\n \Rightarrow -4c = 44 \Rightarrow c = -11
\n $$", "So the full quadratic equation is:
\n $$
\n x^2 + 2x - 11 = 0
\n $$", "2. Apply the quadratic formula:
\n $$
\n x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-11)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 44}}{2} = \frac{-2 \pm \sqrt{48}}{2}
\n $$", "3. Simplify the square root:
\n $$
\n \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}
\n $$", "So:
\n $$
\n x = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3}
\n $$", "---", "### Final Answer", "The two solutions to the equation $ x^2 + 2x - 11 = 0 $ are:", "$$
\nx = -1 + 2\sqrt{3} \quad \ ext{and} \quad x = -1 - 2\sqrt{3}
\n$$", "---", "### Why Use the Quadratic Formula?", "- Efficiently solves any quadratic equation without factoring.
\n- Works even when the equation is not easily factorable.
\n- Helps understand the nature of roots via the discriminant ($ b^2 - 4ac $).
\n- Enables quick calculation in physics, engineering, economics, and beyond.", "---", "### When to Use This Formula", "Use the quadratic formula whenever a quadratic equation is in standard form $ ax^2 + bx + c = 0 $ and factoring proves difficult or impossible. The example $ x = \frac{-2 \pm \sqrt{4 + 44}}{2} $ teaches pattern recognition and substitution — key skills in algebra.", "---", "### Summary", "Mastering the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ empowers you to solve quadratic equations confidently and accurately. From solving $ x^2 + 2x - 11 = 0 $ to exploring real-world applications, this tool is essential for math learners at all levels.", "Tip: Practice rewriting expressions to match $ ax^2 + bx + c = 0 $ to accurately apply the formula — clarity is key!", "---", "Keywords: quadratic formula, solving quadratics, x = [−2 ± √(4 + 44)] / 2, solving quadratic equations, discriminant, algebra tutorial, math help, quadratic formula steps, real-world math applications"]