x^2 = 14 + 2\sqrt{7^2 - (2\sqrt{10})^2} = 14 + 2\sqrt{49 - 40} = 14 + 2\sqrt{9} = 14 + 2 \times 3 = 14 + 6 = 20

["Understanding How to Solve the Equation x² = 14 + 2√(7² − (2√10)²) – Step-by-Step Calculation and Explanation", "Equation solving is a cornerstone of algebra, and sometimes expressions involve nested square roots that can seem challenging at first glance. A compelling example is the expression:", "$$\nx^2 = 14 + 2\sqrt{7^2 - (2\sqrt{10})^2}\n$$", "At first, this equation might appear intricate, but breaking it down step-by-step reveals a clear path to the solution. This article demystifies the solution process, illustrating how algebraic manipulation, the properties of square roots, and squaring techniques simplify even the most complex expressions.", "---", "### Step-by-Step Breakdown of the Equation", "Step 1: Simplify the radicals inside the square root", "Begin by evaluating the expression under the radical:", "$$\n\sqrt{7^2 - (2\sqrt{10})^2}\n$$", "Start with computing each component:", "- (7^2 = 49)\n- ((2\sqrt{10})^2 = 2^2 \ imes (\sqrt{10})^2 = 4 \ imes 10 = 40)", "Substitute these values into the square root:", "$$\n\sqrt{49 - 40} = \sqrt{9} = 3\n$$", "---", "Step 2: Substitute back into the original equation", "Now plug the simplified radical back into the equation:", "$$\nx^2 = 14 + 2 \ imes 3 = 14 + 6 = 20\n$$", "---", "Step 3: Solve for (x)", "Since (x^2 = 20), take the square root of both sides to solve for (x):", "$$\nx = \pm\sqrt{20} = \pm\sqrt{4 \ imes 5} = \pm 2\sqrt{5}\n$$", "---", "### Why This Equation Matters", "This problem exemplifies how algebraic expressions involving nested radicals can be simplified systematically. It demonstrates key concepts such as:", "- Simplifying expressions using exponent rules (e.g., ( (2\sqrt{10})^2 = 2^2 \cdot (\sqrt{10})^2 ))\n- Evaluating square roots stepwise to avoid complexity\n- Applying basic arithmetic to equations with radicals\n- Solving quadratic forms even when not explicitly presented as a standard equation", "---", "### Practical Applications", "While this particular equation may appear abstract, similar algebraic manipulations appear in:", "- Physics problems involving motion equations or energy calculations\n- Engineering formulas where square roots model physical phenomena\n- Computer graphics and game development, where distance and scaling calculations use square roots", "Understanding how to dissect such expressions enables deeper problem-solving skills across STEM disciplines.", "---", "### Final Summary", "- Start with simplifying expressions inside radicals\n- Use exponent and root properties to reduce complexity\n- Perform arithmetic carefully under the radical and afterward\n- Recognize that solving (x^2 = k) yields two solutions: (x = \pm\sqrt{k})", "---", "Final Result:\n$$\nx^2 = 14 + 2\sqrt{7^2 - (2\sqrt{10})^2} \Rightarrow x^2 = 20 \Rightarrow x = \pm 2\sqrt{5}\n$$", "Harnessing stepwise simplification empowers learners to confidently tackle increasingly complex algebra—no nested radical too difficult!", "---", "Keywords: algebra handy, solving x² equation, simplifying square roots, radical expressions, step-by-step algebra, solving quadratic radicals, mathematical problem solving, learn algebra tricks, equation solving tutorial", "---", "If you enjoy breaking down math mysteries, practice more nested radicals—each one strengthens your algebraic intuition!"]