Equation: (12 + 2x)(8 + 2x) = 200

["Equation: (12 + 2x)(8 + 2x) = 200 – Solve Step-by-Step", "If you're tackling algebra, equations like (12 + 2x)(8 + 2x) = 200 can seem tricky at first—but with the right approach, solving them becomes manageable and even satisfying. Whether you're a student preparing for exams, a teacher explaining quadratics, or someone brushing up on algebraic techniques, this article guides you through solving and understanding this quadratic equation.", "---", "### What Is the Equation?", "The equation:\n(12 + 2x)(8 + 2x) = 200\nis a product of two linear binomials set equal to a constant. Expanding this expression reveals a quadratic equation, which can be solved by simplifying, rearranging terms, and factoring or applying the quadratic formula.", "---", "### Why Solve This Type of Equation?", "Understanding how to solve equations like (12 + 2x)(8 + 2x) = 200 helps strengthen skills in:", "- Expanding binomials\n- Simplifying expressions\n- Recognizing quadratic forms\n- Applying algebraic identities\n- Applying substitution techniques", "Plus, equations with real-world modeling often boil down to this algebraic form.", "---", "### Step-by-Step Solution", "#### Step 1: Expand the Left Side", "Start by expanding the product (12 + 2x)(8 + 2x) using the distributive property (FOIL method):", "[\n(12 + 2x)(8 + 2x) = 12 \cdot 8 + 12 \cdot 2x + 2x \cdot 8 + 2x \cdot 2x\n]", "Calculate each term:", "- (12 \cdot 8 = 96)\n- (12 \cdot 2x = 24x)\n- (2x \cdot 8 = 16x)\n- (2x \cdot 2x = 4x^2)", "Add them together:", "[\n96 + 24x + 16x + 4x^2 = 4x^2 + 40x + 96\n]", "So the equation becomes:", "[\n4x^2 + 40x + 96 = 200\n]", "---", "#### Step 2: Bring All Terms to One Side", "Subtract 200 from both sides to form a standard quadratic equation:", "[\n4x^2 + 40x + 96 - 200 = 0 \quad \Rightarrow \quad 4x^2 + 40x - 104 = 0\n]", "---", "#### Step 3: Simplify the Equation", "Divide the entire equation by the greatest common divisor (GCD) of the coefficients to simplify. Here, all coefficients are divisible by 4:", "[\nx^2 + 10x - 26 = 0\n]", "Now you can work with this simplified quadratic:", "[\nx^2 + 10x - 26 = 0\n]", "---", "#### Step 4: Solve the Quadratic Equation", "Two common methods to solve quadratic equations:", "Method A: Factoring (if possible)\nTry to factor (x^2 + 10x - 26 = 0). Look for two numbers that multiply to -26 and add to 10.\nUnfortunately, no simple integer pairs satisfy both conditions. So, factoring is not straightforward here.", "Method B: Use the Quadratic Formula\nFor any quadratic equation (ax^2 + bx + c = 0), the solutions are:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plug in (a = 1), (b = 10), (c = -26):", "[\nx = \frac{-10 \pm \sqrt{(10)^2 - 4(1)(-26)}}{2(1)} = \frac{-10 \pm \sqrt{100 + 104}}{2} = \frac{-10 \pm \sqrt{204}}{2}\n]", "Simplify (\sqrt{204}):\n(204 = 4 \cdot 51), so (\sqrt{204} = 2\sqrt{51})", "[\nx = \frac{-10 \pm 2\sqrt{51}}{2} = -5 \pm \sqrt{51}\n]", "---", "### Step 5: Final Solutions", "[\nx = -5 + \sqrt{51} \quad \ ext{or} \quad x = -5 - \sqrt{51}\n]", "---", "### Bonus Tips", "- Always verify solutions by plugging them back into the original equation.\n- Use a calculator for square roots when needed—ensure accuracy.\n- Recognizing when to simplify first (like dividing by 4 here) saves time.\n- Understanding the concept behind the formula improves flexibility when equations resist simple factoring.", "---", "### Summary", "The equation (12 + 2x)(8 + 2x) = 200 expands into a quadratic:\n[\n4x^2 + 40x - 104 = 0 \quad \ ext{or simplified} \quad x^2 + 10x - 26 = 0\n]", "Solving gives two real solutions:\n[\nx = -5 \pm \sqrt{51}\n]", "Mastering these steps builds a strong foundation for algebra and higher math. Keep practicing expansions, simplifications, and quadratic solving—you’ll gain confidence fast!", "---", "Related searches:\n- How to solve (a+bx)(c+dx)=k\n- Quadratic equation solutions step-by-step\n- Simplify and solve linear and quadratic expressions\n- Algebraic equations for students 2024\n- Expand and solve product of binomials", "---", "Keywords: Equation (12 + 2x)(8 + 2x) = 200, quadratic equation solution, algebra step-by-step, solve quadratic by expanding, linear binomial product, simplified quadratic formula, real solutions to quadratics, high school algebra, algebra tutorial, factoring and quadratic formula."]