A rectangular yard has a length of (x+2) m and a width of (x-3) m. If the area of the yard is 40 m², find the value of x.

You're on a roll — let's solve this geometry problem. Step 1: Set up the equation for the area of the rectangular yard. The length (L) of the yard is (x+2) m and the width (W) is (x-3) m. The area (A) is given as 40 m^2. The formula for the area of a rectangle is A = L × W. Substitute the given expressions into the formula: (x+2)(x-3) = 40 Step 2: Expand and simplify the equation. Multiply the binomials: x^2 - 3x + 2x - 6 = 40 x^2 - x - 6 = 40 Step 3: Rearrange the equation into standard quadratic form ax^2 + bx + c = 0. Subtract 40 from both sides: x^2 - x - 6 - 40 = 0 x^2 - x - 46 = 0 Step 4: Solve the quadratic equation for x using the quadratic formula. The quadratic formula is x = -b ± sqrt(b^2 - 4ac)2a. For x^2 - x - 46 = 0, we have a=1, b=-1, and c=-46. x = -(-1) ± sqrt((-1)^2 - 4(1)(-46))2(1) x = 1 ± sqrt(1 + 184)2 x = 1 ± sqrt(185)2 Step 5: Determine the valid value for x. We have two possible values for x: x_1 = 1 + sqrt(185)2 and x_2 = 1 - sqrt(185)2. Since sqrt(185) ≈ 13.60, let's evaluate these: x_1 ≈ (1 + 13.60)/(2) = (14.60)/(2) = 7.30 x_2 ≈ (1 - 13.60)/(2) = (-12.60)/(2) = -6.30 The width of the yard is (x-3) m. If we use x = -6.30, the width would be -6.30 - 3 = -9.30 m, which is not possible for a physical dimension. Therefore, we must use the positive value for x: x = 1 + sqrt(185)2 Step 6: Calculate the length and width of the yard. Substitute the valid value of x back into the expressions for length and width: Length L = x+2 = (1 + sqrt(185)2) + 2 = 1 + sqrt(185) + 42 = 5 + sqrt(185)2 m Width W = x-3 = (1 + sqrt(185)2) - 3 = 1 + sqrt(185) - 62 = -5 + sqrt(185)2 m The length and width of the yard are: Length: 5 + sqrt(185)2 m Width: -5 + sqrt(185)2 m Send me the next one 📸