\( x(x+2) = 168 \) → \( x^2 + 2x - 168 = 0 \).

["### How to Solve ( x(x+2) = 168 ): Step-by-Step Solution", "Finding solutions to equations involving expressions like ( x(x+2) = 168 ) begins with transforming them into standard quadratic form. In this article, we explore how to convert and solve the equation ( x(x+2) = 168 ) into ( x^2 + 2x - 168 = 0 ), and walk through the step-by-step process to solve the resulting quadratic equation — ideal for students learning algebra and solving quadratic equations.", "---", "#### Understanding the Original Equation", "The equation given is:", "[\nx(x + 2) = 168\n]", "This expression represents multiplying ( x ) by ( x + 2 ), and setting that product equal to 168. To simplify, we expand the left-hand side:", "[\nx(x + 2) = x^2 + 2x\n]", "So substituting back, the equation becomes:", "[\nx^2 + 2x = 168\n]", "---", "#### Converting to Quadratic Form", "To write this in standard quadratic form ( ax^2 + bx + c = 0 ), subtract 168 from both sides:", "[\nx^2 + 2x - 168 = 0\n]", "Now the equation is simplified and ready to solve using familiar methods like factoring, completing the square, or the quadratic formula.", "---", "#### Solving the Quadratic Equation", "We want to solve:", "[\nx^2 + 2x - 168 = 0\n]", "##### Option 1: Factoring", "We look for two numbers that multiply to ( -168 ) and add to ( +2 ).", "Factors of 168:\n1 × 168, 2 × 84, 3 × 56, 4 × 42, 6 × 28, 7 × 24, 8 × 21, 12 × 14", "Notice ( 14 ) and ( -12 ) multiply to ( -168 ) and add to ( 2 ):", "[\nx^2 + 14x - 12x - 168 = 0\n]", "Group the terms:", "[\n(x^2 + 14x) - (12x + 168) = 0\n]", "Factor each group:", "[\nx(x + 14) - 12(x + 14) = 0\n]", "Factor out ( (x + 14) ):", "[\n(x + 14)(x - 12) = 0\n]", "Set each factor equal to zero:", "[\nx + 14 = 0 \quad \ ext{or} \quad x - 12 = 0\n]", "So:", "[\nx = -14 \quad \ ext{or} \quad x = 12\n]", "##### Option 2: Using the Quadratic Formula (Quick Check)", "The quadratic formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For ( a = 1 ), ( b = 2 ), ( c = -168 ):", "[\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-168)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 672}}{2} = \frac{-2 \pm \sqrt{676}}{2}\n]", "[\n\sqrt{676} = 26\n]", "[\nx = \frac{-2 \pm 26}{2}\n]", "[\nx = \frac{24}{2} = 12 \quad \ ext{or} \quad x = \frac{-28}{2} = -14\n]", "Same solutions confirmed.", "---", "#### Why This Equation Matters", "This type of equation appears in real-world contexts such as geometry problems, profit calculations, and physics problems involving motion or area optimization. Converting expressions like ( x(x+2) ) into a quadratic form is a foundational skill for modeling and solving complex word problems.", "---", "#### Summary", "To convert and solve ( x(x+2) = 168 ):", "1. Expand to get ( x^2 + 2x = 168 )\n2. Rewrite as ( x^2 + 2x - 168 = 0 )\n3. Factor or apply the quadratic formula to find solutions\n4. Confirm both ( x = 12 ) and ( x = -14 ) are valid", "If you're learning algebra, recognizing how to transform expressions into standard quadratic form unlocks powerful problem-solving tools.", "---", "Keywords: solve ( x(x+2) = 168 ), quadratic equation ( x^2 + 2x - 168 = 0 ), step-by-step quadratic solution, algebra simplification, quadratic equations intro, solve quadratics by factoring, quadratic formula explanation", "---", "Author’s Note: Practice solving quadratic equations daily — combining expansion, simplification, and factoring — accelerates mastery. With every equation you solve, you strengthen your algebra foundation!", "---", "If you want, check both solutions in the original equation to ensure correctness, and explore related problems involving binomial expansions and quadratic models."]