\( x^2 = 4x - x^2 \Rightarrow 2x^2 - 4x = 0 \Rightarrow 2x(x - 2) = 0 \).
["# Solving the Quadratic Equation ( x^2 = 4x - x^2 ): Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, widely used in science, engineering, economics, and everyday problem-solving. One classic equation that demonstrates important principles of algebraic manipulation is:", "[\nx^2 = 4x - x^2\n]", "In this article, we will walk through solving this equation step-by-step, explore how it transforms into ( 2x^2 - 4x = 0 ), and arrive at the final factorized form ( 2x(x - 2) = 0 ). This process not only gives the solution but also strengthens your understanding of equation balance, factoring techniques, and the zero product property.", "---", "### Step 1: Rearranging the Equation", "Start with the original equation:", "[\nx^2 = 4x - x^2\n]", "To bring all terms to one side and form a standard quadratic equation, add ( x^2 ) to both sides:", "[\nx^2 + x^2 = 4x\n]", "[\n2x^2 = 4x\n]", "Now, subtract ( 4x ) from both sides to get zero on the right:", "[\n2x^2 - 4x = 0\n]", "This step is crucial—it transforms the equation into a standard quadratic form ( ax^2 + bx + c = 0 ), where ( a = 2 ), ( b = -4 ), and ( c = 0 ).", "---", "### Step 2: Factoring the Quadratic Expression", "We now solve:", "[\n2x^2 - 4x = 0\n]", "Factor out the greatest common factor. Here, both terms have a factor of ( 2x ):", "[\n2x(x - 2) = 0\n]", "This factoring relies on recognizing distributive properties and extracting common terms. The expression inside the parentheses simplifies neatly to ( x - 2 ), confirming the equivalent nature of both forms.", "---", "### Step 3: Apply the Zero Product Property", "The equation ( 2x(x - 2) = 0 ) applies the zero product property, which states that if a product of factors equals zero, then at least one of the factors must be zero.", "Set each factor equal to zero:", "1. ( 2x = 0 \Rightarrow x = 0 )\n2. ( x - 2 = 0 \Rightarrow x = 2 )", "---", "### Final Answer", "The solutions to the equation ( x^2 = 4x - x^2 ) are:", "[\n\boxed{x = 0 \quad \ ext{and} \quad x = 2}\n]", "---", "### Why This Matters", "Understanding how to manipulate quadratic equations builds a solid foundation for:", "- Modeling real-world scenarios (e.g., profit, motion, geometry)\n- Applying techniques like factoring, completing the square, and quadratic formula\n- Developing logical reasoning and algebraic manipulation skills", "Whether you're a student, teacher, or self-learner, mastering this process empowers you to tackle more complex algebraic challenges with confidence.", "---", "If you'd like to explore related topics such as graphing solutions or applications of the quadratic formula, feel free to check out our other articles!", "---", "Keywords for SEO:\nquadratic equation solution, solve ( x^2 = 4x - x^2 ), factoring ( 2x^2 - 4x = 0 ), factoring technique, zero product property, algebra tutorial, solve quadratic equations step-by-step, real solutions to quadratic equations"]
