The perimeter of a rectangular plot of land is 84 metres and its area is 320m^(2). What are the dimensions of the plot? Record your answers from smallest to largest separated by a comma. (ie. -1.1,2.7,9.8 ) If there are no real roots then write 'no solution' in the blank. ◻

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Perimeter Equation: Let's denote the length of the rectangle as L and the width as W. The perimeter P of a rectangle is given by P = 2(L + W). We are given that the perimeter is 84 meters. So, we have the equation: 2(L + W) = 84Solve for L + W: Divide both sides of the equation by 2 to solve for L + W: L + W = 42Area Equation: We are also given that the area A of the rectangle is 320 square meters. The area of a rectangle is given by A = L * W. So, we have the equation: L * W = 320Express W in terms of L: Now we have a system of two equations: 1) L + W = 42 2) L * W = 320 We can express W in terms of L from the first equation: W = 42 - LQuadratic Equation in L: Substitute W = 42 - L into the second equation: L * (42 - L) = 320 Expand the equation: L^2 - 42L + 320 = 0Quadratic Formula: Now we have a quadratic equation in terms of L. We can solve for L using the quadratic formula, where a = 1, b = -42, and c = 320. The quadratic formula is given by: L = (-b ± √(b^2 - 4ac)) / (2a)Calculate Discriminant: Calculate the discriminant (b^2 - 4ac): Discriminant = (-42)^2 - 4(1)(320) = 1764 - 1280 = 484Two Real Solutions for L: Since the discriminant is positive, we have two real solutions for L. Calculate the square root of the discriminant: √484 = 22Calculate Square Root: Now, plug the values into the quadratic formula to find the two solutions for L: L = (42 ± 22) / 2Two Possible Values for L: Calculate the two possible values for L: L1 = (42 + 22) / 2 = 64 / 2 = 32 L2 = (42 - 22) / 2 = 20 / 2 = 10Find Corresponding Widths: Now we have two possible lengths for the rectangle: L1 = 32 meters and L2 = 10 meters. Using the equation W = 42 - L, we can find the corresponding widths: W1 = 42 - 32 = 10 meters W2 = 42 - 10 = 32 metersDimensions of the Rectangle: We have found the dimensions of the rectangle: 32 meters by 10 meters. Now let's solve the quadratic equation x^2 + 6x + 17 = 0. We can use the quadratic formula again, where a = 1, b = 6, and c = 17.Solve Quadratic Equation: Calculate the discriminant for the quadratic equation: Discriminant = (6)^2 - 4(1)(17) = 36 - 68 = -32Calculate Discriminant: Since the discriminant is negative, there are no real roots for the quadratic equation x^2 + 6x + 17 = 0.

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