This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Form the quadratic equation. The area of a rectangle is given by the formula A = L × W. Given: Length L = (x+4) m Width W = (x-3) m Area A = 77 m^2 Substitute these values into the area formula: (x+4)(x-3) = 77 Expand the left side of the equation: x^2 - 3x + 4x - 12 = 77 x^2 + x - 12 = 77 Rearrange the equation into the standard quadratic form ax^2 + bx + c = 0: x^2 + x - 12 - 77 = 0 x^2 + x - 89 = 0 Step 2: Solve the quadratic equation for x. The quadratic equation is x^2 + x - 89 = 0. We can use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a=1, b=1, and c=-89. x = -1 ± sqrt(1^2 - 4(1)(-89))2(1) x = -1 ± sqrt(1 + 356)2 x = -1 ± sqrt(357)2 Calculate the value of sqrt(357): sqrt(357) ≈ 18.894 Now find the two possible values for x: x_1 = (-1 + 18.894)/(2) = (17.894)/(2) = 8.947 x_2 = (-1 - 18.894)/(2) = (-19.894)/(2) = -9.947 Since x represents a dimension, it must be positive. Therefore, x ≈ 8.947. Step 3: Find the dimensions of the hall. Length L = x+4 L = 8.947 + 4 = 12.947 m Width W = x-3 W = 8.947 - 3 = 5.947 m The dimensions of the hall are approximately: Length: 12.95 m Width: 5.95 m That's 2 down. 3 left today — send the next one.