Expand: \( x^2 + x^2 + 2x + 1 = 145 \) → \( 2x^2 + 2x + 1 = 145 \)

["# Expand ( x^2 + x^2 + 2x + 1 = 145 ) to ( 2x^2 + 2x + 1 = 145 ): A Step-by-Step Guide", "Solving quadratic equations often begins with the symmetric expansion and simplification of expressions. One common transformation involves combining like terms—particularly in equations like ( x^2 + x^2 + 2x + 1 = 145 ). Understanding how to expand and simplify such expressions is essential for effectively solving quadratic equations. In this article, we’ll explore how expanding ( x^2 + x^2 + 2x + 1 = 145 ) leads to ( 2x^2 + 2x + 1 = 145 ), and why this simplification improves clarity and solution paths.", "---", "## Why Combine Like Terms?", "The equation ( x^2 + x^2 + 2x + 1 = 145 ) contains two ( x^2 ) terms, which together combine into ( 2x^2 ). This process reflects a fundamental algebraic principle: like terms combine through addition or subtraction. In this case, the two ( x^2 ) terms are identical and sum neatly to form a polynomial with clearer structure.", "Before expanding, the equation is:\n[\nx^2 + x^2 + 2x + 1 = 145\n]", "Combining the ( x^2 ) terms gives:\n[\n2x^2 + 2x + 1 = 145\n]", "This simplified form makes it easier to isolate ( x ), apply factoring techniques, or use the quadratic formula effectively.", "---", "## Step-by-Step Solution After Simplification", "### Step 1: Bring all terms to one side\nTo solve ( 2x^2 + 2x + 1 = 145 ), subtract 145 from both sides:\n[\n2x^2 + 2x + 1 - 145 = 0\n]\n[\n2x^2 + 2x - 144 = 0\n]", "---", "### Step 2: Simplify the quadratic equation", "Divide every term by 2 to reduce coefficients:\n[\nx^2 + x - 72 = 0\n]", "This simplified equation is now ready for factoring, completing the square, or applying the quadratic formula.", "---", "### Step 3: Factor the quadratic", "We look for two numbers that multiply to ( -72 ) and add to ( 1 ). These numbers are ( 9 ) and ( -8 ):\n[\n(x + 9)(x - 8) = 0\n]", "---", "### Step 4: Solve for ( x )", "Set each factor equal to zero:\n[\nx + 9 = 0 \quad \Rightarrow \quad x = -9\n]\n[\nx - 8 = 0 \quad \Rightarrow \quad x = 8\n]", "---", "## Final Answer", "The solutions to the original equation ( x^2 + x^2 + 2x + 1 = 145 ) are:\n[\n\boxed{x = -9 \quad} \ ext{and} \quad x = 8\n]", "---", "## Key Takeaways", "- Combining like terms — such as merging ( x^2 + x^2 ) into ( 2x^2 ) — streamlines equations and reduces errors.\n- Always bring all constant terms to one side to form a standard quadratic equation.\n- Simplifying before applying formulas like the quadratic formula improves accuracy and interpretation.", "Mastering these expansion and simplification techniques empowers you to tackle more complex algebra problems with confidence and precision.", "---", "Keywords: expand ( x^2 + x^2 + 2x + 1 = 145 ), simplify quadratic equations, solve ( 2x^2 + 2x + 1 = 145 ), algebraic simplification, quadratic equation solutions, combine like terms, algebraic techniques."]