\[ x^2 = 8 + 4 = 12 \Rightarrow x = \pm \sqrt{12} = \pm 2\sqrt{3}. \]

["# Solving ( x^2 = 8 + 4 ): A Simple Guide to Finding ( x = \pm 2\sqrt{3} )", "If you’ve ever seen the equation\n[ x^2 = 8 + 4 ]\nand wondered how to solve for ( x ), you’re in the right place! This article breaks down the steps clearly, explaining everything from simplifying the right side to isolating ( x ), with a focus on understanding the final solution:\n[ x = \pm \sqrt{12} = \pm 2\sqrt{3} ]", "---", "## Step 1: Simplify the Right Side", "Start by simplifying the expression on the right-hand side of the equation:\n[ x^2 = 8 + 4 = 12 ]", "So:\n[ x^2 = 12 ]", "This simplification is crucial because it reduces the problem to a straightforward square root operation.", "---", "## Step 2: Take the Square Root of Both Sides", "To solve for ( x ), we take the square root of both sides:\n[ x = \pm \sqrt{12} ]\n(Note: When solving ( x^2 = a ), the solution is ( x = \pm \sqrt{a} ))", "---", "## Step 3: Simplify ( \sqrt{12} )", "The square root of 12 isn’t a clean whole number, so we simplify it:\n[ \sqrt{12} = \sqrt{4 \ imes 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} ]", "Thus:\n[ x = \pm \sqrt{12} = \pm 2\sqrt{3} ]", "---", "## Why This Matters: Understanding the Solution Set", "Since squaring both positive and negative numbers yields the same result (e.g., ( (2\sqrt{3})^2 = (-2\sqrt{3})^2 = 12 )), there are two real solutions:\n[ x = 2\sqrt{3} \quad \ ext{and} \quad x = -2\sqrt{3} ]", "These are known as the root solutions of the equation.", "---", "## Final Answer", "[ \boxed{x = \pm 2\sqrt{3}} ]", "---", "## Key Takeaways", "- Simplify first: Always simplify constants before applying root operations.\n- Square roots have positive and negative solutions: ( x^2 = a \Rightarrow x = \pm \sqrt{a} )\n- Rationalize radicals: Express square roots in simplest radical form for clarity.", "---", "## Additional Tips for Solving Quadratic Equations", "This foundational method applies to more complex equations:\n- Combine like terms before taking roots\n- Remember both ( + ) and ( - ) signs in solutions\n- Simplify radicals to make answers cleaner and more understandable", "Mastering this process helps with everything from basic algebra to advanced math topics—so keep practicing!", "---", "Keywords: solve ( x^2 = 8 + 4 ), ( x = \pm \sqrt{12} ), simplify ( \sqrt{12} ), rationalizing square roots, algebraic equations, step-by-step solution, positive and negative roots, ( x = \pm 2\sqrt{3} )", "---", "Image suggestion: A diagram showing ( x^2 = 12 ) with arrows indicating the two solution branches: ( x = 2\sqrt{3} ) and ( x = -2\sqrt{3} )", "---", "If you found this guide helpful, share it to master solving quadratic equations! 💡"]