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.
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Here's the solution to the quadratic equations problem: Part (a): Form a quadratic equation representing the situation. Step 1: Define variables. Let the width of the rectangular playground be w meters. The length of the playground is 5 meters more than its width, so the length l = w + 5 meters. The area of the playground is given as 600\,m^2. Step 2: Form the equation. The area of a rectangle is given by the formula Area = length × width. Substitute the expressions for length, width, and the given area: 600 = (w + 5) × w 600 = w^2 + 5w Rearrange the equation into the standard quadratic form ax^2 + bx + c = 0: w^2 + 5w - 600 = 0 The quadratic equation representing the situation is w^2 + 5w - 600 = 0. Part (b): Solve the equation by completing the square. Step 1: Isolate the variable terms. Start with the equation from part (a): w^2 + 5w - 600 = 0 Move the constant term to the right side of the equation: w^2 + 5w = 600 Step 2: Complete the square. To complete the square for w^2 + 5w, take half of the coefficient of w (which is 5), square it, and add it to both sides of the equation. Half of 5 is (5)/(2). Squaring it gives ((5)/(2))^2 = (25)/(4). w^2 + 5w + (25)/(4) = 600 + (25)/(4) Step 3: Factor the left side and simplify the right side. The left side is now a perfect square trinomial: (w + (5)/(2))^2 = (2400)/(4) + (25)/(4) (w + (5)/(2))^2 = (2425)/(4) Step 4: Take the square root of both sides. w + (5)/(2) = ±sqrt((2425)/(4)) w + (5)/(2) = ±sqrt(2425)2 Note that sqrt(2425) = sqrt(25 × 97) = 5sqrt(97). w + (5)/(2) = ±5sqrt(97)2 Step 5: Solve for w. Subtract (5)/(2) from both sides: w = -(5)/(2) ± 5sqrt(97)2 w = -5 ± 5sqrt(97)2 The solutions to the equation are w = -5 + 5sqrt(97)2 or w = -5 - 5sqrt(97)2. Part (c): Find the dimensions of the playground. Step 1: Choose the valid value for width. From part (b), we have two possible values for w: w_1 = -5 + 5sqrt(97)2 w_2 = -5 - 5sqrt(97)2 Since width cannot be a negative value, we must choose the positive solution. sqrt(97) ≈ 9.8488. w_1 = (-5 + 5(9.8488))/(2) = (-5 + 49.244)/(2) = (44.244)/(2) ≈ 22.122\,m. w_2 = (-5 - 5(9.8488))/(2) = (-5 - 49.244)/(2) = (-54.244)/(2) ≈ -27.122\,m. Therefore, the width w = -5 + 5sqrt(97)2\,m. Step 2: Calculate the length. The length l = w + 5. l = (-5 + 5sqrt(97)2) + 5 To add 5, express it with a denominator of 2: 5 = (10)/(2). l = -5 + 5sqrt(97)2 + (10)/(2) l = -5 + 5sqrt(97) + 102 l = 5 + 5sqrt(97)2\,m Step 3: State the dimensions. The dimensions of the playground are: Width: -5 + 5sqrt(97)2\,m Length: 5 + 5sqrt(97)2\,m You've completed all your questions for today! Great work.