Expanding: 2w^2 - 2w + 6w - 6 = 24 → 2w^2 + 4w - 30 = 0.

["Expanding and Simplifying Quadratic Equations: Solving (2w^2 - 2w + 6w - 6 = 24) Step-by-Step", "Expanding and simplifying algebraic expressions is a fundamental skill in algebra, crucial for solving equations and understanding quadratic relationships. One common task involves manipulating expressions to standard quadratic form—essential when applying techniques like factoring, completing the square, or using the quadratic formula. In this article, we’ll expand and simplify the equation:", "[\n2w^2 - 2w + 6w - 6 = 24\n]", "to the form:", "[\n24 \ %\ (2w^2 + 4w - 30) = 0\n]", "and explain the logical steps behind the transformation.", "---", "### Step 1: Simplify the Left-Hand Side", "Start by combining like terms on the left-hand side of the equation:", "[\n2w^2 - 2w + 6w - 6\n]", "Combine the linear ((w)) terms:\n[\n-2w + 6w = 4w\n]", "So the simplified left side becomes:\n[\n2w^2 + 4w - 6\n]", "Now the equation is:\n[\n2w^2 + 4w - 6 = 24\n]", "---", "### Step 2: Move All Terms to One Side to Form Standard Quadratic Form", "To convert the equation into standard quadratic form ((ax^2 + bx + c = 0)), subtract 24 from both sides:", "[\n2w^2 + 4w - 6 - 24 = 0\n]", "Simplify the constants:\n[\n2w^2 + 4w - 30 = 0\n]", "This matches the target form:\n[\n24 \ %\ (2w^2 + 4w - 30) = 0\n]", "Note that "24 % " in the original expression indicates subtraction of 24, hence the left-hand side becomes (2w^2 + 4w - 30).", "---", "### Why Expand and Simplify?", "Although the equation was initially on one side, expanding and simplifying allows for clearer analysis and easier substitution into solving tools. The transformed form:", "[\n24 \ %\ (2w^2 + 4w - 30) = 0\n]", "explicitly shows the entire quadratic expression after moving all terms to one side, making it ready for root-finding methods. It confirms the equivalent expression:", "[\n2w^2 + 4w - 30 = 0\n]", "---", "### Step 3: Standard Quadratic Equation", "We now have:\n[\n2w^2 + 4w - 30 = 0\n]", "Divide through by 2 to simplify:\n[\nw^2 + 2w - 15 = 0\n]", "This is now in standard quadratic form, ideal for factoring, completing the square, or applying the quadratic formula.", "---", "### Step 4: Solving the Quadratic Equation", "Factor the quadratic:\n[\nw^2 + 2w - 15 = (w + 5)(w - 3) = 0\n]", "Set each factor equal to zero:\n[\nw + 5 = 0 \quad \Rightarrow \quad w = -5\n]\n[\nw - 3 = 0 \quad \Rightarrow \quad w = 3\n]", "---", "### Conclusion: Key Takeaways", "Expanding and simplifying algebraic expressions—particularly moving all terms to one side—creates clarity and prepares equations for solution. The transformation\n[\n2w^2 - 2w + 6w - 6 = 24 \quad \ o \quad 24 \ % \ (2w^2 + 4w - 30) = 0\n]\nis critical for accessing the true quadratic form (2w^2 + 4w - 30 = 0). Mastering such steps strengthens equation-solving proficiency, supports algebraic mastery, and builds confidence for advanced math challenges.", "---", "Keywords: expand quadratic equation, simplify (2w^2 - 2w + 6w - 6 = 24), standard quadratic form, solving quadratics, algebraic manipulation, factoring quadratic, quadratic formula, algebra tips, educational algebra guide.", "Meta Description:\nLearn how to expand and simplify (2w^2 - 2w + 6w - 6 = 24) to standard quadratic form (2w^2 + 4w - 30 = 0). Step-by-step explanation with factoring and solution. # algebra #quadratic_equations #math_tips", "---", "Understanding how to manipulate and simplify quadratic expressions like (2w^2 - 2w + 6w - 6 = 24) opens the door to solving real-world problems in physics, engineering, and economics where quadratic relationships emerge."]