A rectangular solar panel has length \(2x + 3\) meters and width \(x - 1\) meters. If the area is 40 m², find \(x\).
["How to Find the Value of ( x ) for a Rectangular Solar Panel with Area 40 m²", "Solar panel efficiency and space optimization are key in renewable energy design. Understanding how to calculate the dimensions of solar panels using algebra helps both engineers and homeowners maximize usage of available space. This article explains how to find the value of ( x ) when a rectangular solar panel has length ( 2x + 3 ) meters, width ( x - 1 ) meters, and an area of 40 m².", "---", "### Given Dimensions and Area", "- Length: ( 2x + 3 ) meters\n- Width: ( x - 1 ) meters\n- Area: 40 m²", "Since area of a rectangle is length multiplied by width, we write the equation:", "[\n(2x + 3)(x - 1) = 40\n]", "---", "### Step 1: Expand the Left-Hand Side", "Use the distributive property (FOIL method) to expand:", "[\n(2x + 3)(x - 1) = 2x \cdot x + 2x \cdot (-1) + 3 \cdot x + 3 \cdot (-1)\n]", "[\n= 2x^2 - 2x + 3x - 3\n]", "Combine like terms:", "[\n2x^2 + x - 3\n]", "So the equation becomes:", "[\n2x^2 + x - 3 = 40\n]", "---", "### Step 2: Move All Terms to One Side", "Subtract 40 from both sides to form a quadratic equation:", "[\n2x^2 + x - 3 - 40 = 0\n]", "[\n2x^2 + x - 43 = 0\n]", "---", "### Step 3: Solve the Quadratic Equation", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 2 ), ( b = 1 ), and ( c = -43 ). Plug in the values:", "[\nx = \frac{-1 \pm \sqrt{(1)^2 - 4(2)(-43)}}{2(2)}\n]", "[\nx = \frac{-1 \pm \sqrt{1 + 344}}{4}\n]", "[\nx = \frac{-1 \pm \sqrt{345}}{4}\n]", "Since ( \sqrt{345} ) is irrational (approximately 18.57), we keep it exact:", "[\nx = \frac{-1 \pm \sqrt{345}}{4}\n]", "Only the positive root makes sense in a real-world context (since ( x - 1 ) must be positive for a valid width):", "[\nx = \frac{-1 + \sqrt{345}}{4}\n]", "---", "### Step 4: Approximate the Value (Optional for Clarity)", "[\n\sqrt{345} \approx 18.57\n]", "[\nx \approx \frac{-1 + 18.57}{4} = \frac{17.57}{4} \approx 4.39\n]", "---", "### Final Answer", "The value of ( x ) that satisfies the area equation is:", "[\n\boxed{x = \frac{-1 + \sqrt{345}}{4}}\n]", "This value ensures the solar panel has dimensions ( 2x + 3 ) meters by ( x - 1 ) meters and an area exactly equal to 40 m².", "---", "### Why This Matters", "Correctly modeling solar panel geometry aids in maximizing energy output per square meter. Solving for variables algebraically ensures precise design, especially when space is limited or efficiency is critical.", "---", "Keywords: rectangular solar panel area, solve for ( x ), solar panel dimensions, algebraic equation, quadratic formula, solar energy optimization, geometry in renewable energy."]
