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Simplifying (60 + -1d)(40 + -2d) = 1312.5 Multiply (60 + -1d) * (40 + -2d) (60(40 + -2d) + -1d * (40 + -2d)) = 1312.5 ((40 * 60 + -2d * 60) + -1d * (40 + -2d)) = 1312.5 ((2400 + -120d) + -1d * (40 + -2d)) = 1312.5 (2400 + -120d + (40 * -1d + -2d * -1d)) = 1312.5 (2400 + -120d + (-40d + 2d2)) = 1312.5 Combine like terms: -120d + -40d = -160d (2400 + -160d + 2d2) = 1312.5 Solving 2400 + -160d + 2d2 = 1312.5 Solving for variable 'd'. Reorder the terms: 2400 + -1312.5 + -160d + 2d2 = 1312.5 + -1312.5 Combine like terms: 2400 + -1312.5 = 1087.5 1087.5 + -160d + 2d2 = 1312.5 + -1312.5 Combine like terms: 1312.5 + -1312.5 = 0.0 1087.5 + -160d + 2d2 = 0.0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 543.75 + -80d + d2 = 0.0 Move the constant term to the right: Add '-543.75' to each side of the equation. 543.75 + -80d + -543.75 + d2 = 0.0 + -543.75 Reorder the terms: 543.75 + -543.75 + -80d + d2 = 0.0 + -543.75 Combine like terms: 543.75 + -543.75 = 0.00 0.00 + -80d + d2 = 0.0 + -543.75 -80d + d2 = 0.0 + -543.75 Combine like terms: 0.0 + -543.75 = -543.75 -80d + d2 = -543.75 The d term is -80d. Take half its coefficient (-40). Square it (1600) and add it to both sides. Add '1600' to each side of the equation. -80d + 1600 + d2 = -543.75 + 1600 Reorder the terms: 1600 + -80d + d2 = -543.75 + 1600 Combine like terms: -543.75 + 1600 = 1056.25 1600 + -80d + d2 = 1056.25 Factor a perfect square on the left side: (d + -40)(d + -40) = 1056.25 Calculate the square root of the right side: 32.5 Break this problem into two subproblems by setting (d + -40) equal to 32.5 and -32.5.Subproblem 1
d + -40 = 32.5 Simplifying d + -40 = 32.5 Reorder the terms: -40 + d = 32.5 Solving -40 + d = 32.5 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '40' to each side of the equation. -40 + 40 + d = 32.5 + 40 Combine like terms: -40 + 40 = 0 0 + d = 32.5 + 40 d = 32.5 + 40 Combine like terms: 32.5 + 40 = 72.5 d = 72.5 Simplifying d = 72.5Subproblem 2
d + -40 = -32.5 Simplifying d + -40 = -32.5 Reorder the terms: -40 + d = -32.5 Solving -40 + d = -32.5 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '40' to each side of the equation. -40 + 40 + d = -32.5 + 40 Combine like terms: -40 + 40 = 0 0 + d = -32.5 + 40 d = -32.5 + 40 Combine like terms: -32.5 + 40 = 7.5 d = 7.5 Simplifying d = 7.5Solution
The solution to the problem is based on the solutions from the subproblems. d = {72.5, 7.5}
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