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Simplifying 21d2 + -640d + 1728 = 0 Reorder the terms: 1728 + -640d + 21d2 = 0 Solving 1728 + -640d + 21d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. 82.28571429 + -30.47619048d + d2 = 0 Move the constant term to the right: Add '-82.28571429' to each side of the equation. 82.28571429 + -30.47619048d + -82.28571429 + d2 = 0 + -82.28571429 Reorder the terms: 82.28571429 + -82.28571429 + -30.47619048d + d2 = 0 + -82.28571429 Combine like terms: 82.28571429 + -82.28571429 = 0.00000000 0.00000000 + -30.47619048d + d2 = 0 + -82.28571429 -30.47619048d + d2 = 0 + -82.28571429 Combine like terms: 0 + -82.28571429 = -82.28571429 -30.47619048d + d2 = -82.28571429 The d term is -30.47619048d. Take half its coefficient (-15.23809524). Square it (232.1995465) and add it to both sides. Add '232.1995465' to each side of the equation. -30.47619048d + 232.1995465 + d2 = -82.28571429 + 232.1995465 Reorder the terms: 232.1995465 + -30.47619048d + d2 = -82.28571429 + 232.1995465 Combine like terms: -82.28571429 + 232.1995465 = 149.91383221 232.1995465 + -30.47619048d + d2 = 149.91383221 Factor a perfect square on the left side: (d + -15.23809524)(d + -15.23809524) = 149.91383221 Calculate the square root of the right side: 12.243930423 Break this problem into two subproblems by setting (d + -15.23809524) equal to 12.243930423 and -12.243930423.Subproblem 1
d + -15.23809524 = 12.243930423 Simplifying d + -15.23809524 = 12.243930423 Reorder the terms: -15.23809524 + d = 12.243930423 Solving -15.23809524 + d = 12.243930423 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '15.23809524' to each side of the equation. -15.23809524 + 15.23809524 + d = 12.243930423 + 15.23809524 Combine like terms: -15.23809524 + 15.23809524 = 0.00000000 0.00000000 + d = 12.243930423 + 15.23809524 d = 12.243930423 + 15.23809524 Combine like terms: 12.243930423 + 15.23809524 = 27.482025663 d = 27.482025663 Simplifying d = 27.482025663Subproblem 2
d + -15.23809524 = -12.243930423 Simplifying d + -15.23809524 = -12.243930423 Reorder the terms: -15.23809524 + d = -12.243930423 Solving -15.23809524 + d = -12.243930423 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '15.23809524' to each side of the equation. -15.23809524 + 15.23809524 + d = -12.243930423 + 15.23809524 Combine like terms: -15.23809524 + 15.23809524 = 0.00000000 0.00000000 + d = -12.243930423 + 15.23809524 d = -12.243930423 + 15.23809524 Combine like terms: -12.243930423 + 15.23809524 = 2.994164817 d = 2.994164817 Simplifying d = 2.994164817Solution
The solution to the problem is based on the solutions from the subproblems. d = {27.482025663, 2.994164817}
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