Gleichung: \((150 + 2x)(80 + 2x) = 15.120\)

["Solving the Equation: ((150 + 2x)(80 + 2x) = 15.120) – A Step-by-Step Guide", "When faced with an equation like ((150 + 2x)(80 + 2x) = 15.120), solving it algebraically reveals valuable insights into quadratic relationships. This article explains how to expand, simplify, and solve such equations—perfect for students, educators, or math enthusiasts looking to deepen their understanding of quadratic expressions and equation solving.", "---", "### Understanding the Equation", "The equation ((150 + 2x)(80 + 2x) = 15.120) combines two linear expressions into a product equaling a numeric value. Such forms often appear in real-world modeling, such as in area calculations, optimization problems, or financial formulas.", "---", "### Step 1: Expand the Product", "First, expand the left-hand side using distributive property (FOIL method):", "[\n(150 + 2x)(80 + 2x) = 150 \cdot 80 + 150 \cdot 2x + 2x \cdot 80 + 2x \cdot 2x\n]", "Calculate each term:", "- (150 \cdot 80 = 12.000)\n- (150 \cdot 2x = 300x)\n- (2x \cdot 80 = 160x)\n- (2x \cdot 2x = 4x^2)", "Add them together:", "[\n12.000 + 300x + 160x + 4x^2 = 4x^2 + 460x + 12.000\n]", "---", "### Step 2: Write the Quadratic Equation", "Set the expanded expression equal to 15.120:", "[\n4x^2 + 460x + 12.000 = 15.120\n]", "Subtract (15.120) from both sides:", "[\n4x^2 + 460x + 12.000 - 15.120 = 0\n]", "[\n4x^2 + 460x - 3.120 = 0\n]", "---", "### Step 3: Simplify the Equation", "Divide every term by 4 to reduce coefficients:", "[\nx^2 + 115x - 0.78 = 0\n]", "This simplified quadratic equation is easier to solve using standard methods.", "---", "### Step 4: Solve Using the Quadratic Formula", "The quadratic formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, (a = 1), (b = 115), and (c = -0.78).", "Calculate the discriminant:", "[\n\Delta = b^2 - 4ac = 115^2 - 4(1)(-0.78) = 13.225 + 3.12 = 16.345\n]", "Take the square root:", "[\n\sqrt{16.345} \approx 4.043\n]", "Now compute the two solutions:", "[\nx = \frac{-115 \pm 4.043}{2}\n]", "[\nx_1 = \frac{-115 + 4.043}{2} = \frac{-110.957}{2} = -55.4785\n]", "[\nx_2 = \frac{-115 - 4.043}{2} = \frac{-119.043}{2} = -59.5215\n]", "---", "### Step 5: Interpret the Solutions", "Both solutions are negative real numbers, meaning (x \approx -55.48) or (x \approx -59.52). Depending on the problem’s context—such as physical measurements or financial variables—only positive (x) values might make sense; otherwise, the equation could model non-physical or abstract relationships.", "---", "### Why This Equation Matters", "Equations of the form ((a + bx)(c + dx) = k) often model relationships where two linear factors determine a target value. Knowledge of algebraic manipulation and quadratic solving is essential in physics, engineering, and economics for analyzing nonlinear systems.", "---", "### Final Thoughts", "Solving ((150 + 2x)(80 + 2x) = 15.120) teaches core algebraic skills: expansion, simplification, quadratic formula application, and interpretation of results. Whether for homework, test prep, or real-world problem-solving, mastering these steps builds a strong foundation in mathematics.", "---", "Keywords: solution to ((150 + 2x)(80 + 2x) = 15.120), quadratic equation solving, algebraic manipulation, discriminant application, step-by-step math tutorial, real-world equation, quadratic formula, math education", "---", "By understanding and solving such equations, you unlock deeper mathematical reasoning crucial for academic success and practical problem-solving."]