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Simplifying 6d2 + 21d = 10d + 25 Reorder the terms: 21d + 6d2 = 10d + 25 Reorder the terms: 21d + 6d2 = 25 + 10d Solving 21d + 6d2 = 25 + 10d Solving for variable 'd'. Reorder the terms: -25 + 21d + -10d + 6d2 = 25 + 10d + -25 + -10d Combine like terms: 21d + -10d = 11d -25 + 11d + 6d2 = 25 + 10d + -25 + -10d Reorder the terms: -25 + 11d + 6d2 = 25 + -25 + 10d + -10d Combine like terms: 25 + -25 = 0 -25 + 11d + 6d2 = 0 + 10d + -10d -25 + 11d + 6d2 = 10d + -10d Combine like terms: 10d + -10d = 0 -25 + 11d + 6d2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -4.166666667 + 1.833333333d + d2 = 0 Move the constant term to the right: Add '4.166666667' to each side of the equation. -4.166666667 + 1.833333333d + 4.166666667 + d2 = 0 + 4.166666667 Reorder the terms: -4.166666667 + 4.166666667 + 1.833333333d + d2 = 0 + 4.166666667 Combine like terms: -4.166666667 + 4.166666667 = 0.000000000 0.000000000 + 1.833333333d + d2 = 0 + 4.166666667 1.833333333d + d2 = 0 + 4.166666667 Combine like terms: 0 + 4.166666667 = 4.166666667 1.833333333d + d2 = 4.166666667 The d term is 1.833333333d. Take half its coefficient (0.9166666665). Square it (0.8402777775) and add it to both sides. Add '0.8402777775' to each side of the equation. 1.833333333d + 0.8402777775 + d2 = 4.166666667 + 0.8402777775 Reorder the terms: 0.8402777775 + 1.833333333d + d2 = 4.166666667 + 0.8402777775 Combine like terms: 4.166666667 + 0.8402777775 = 5.0069444445 0.8402777775 + 1.833333333d + d2 = 5.0069444445 Factor a perfect square on the left side: (d + 0.9166666665)(d + 0.9166666665) = 5.0069444445 Calculate the square root of the right side: 2.237620264 Break this problem into two subproblems by setting (d + 0.9166666665) equal to 2.237620264 and -2.237620264.Subproblem 1
d + 0.9166666665 = 2.237620264 Simplifying d + 0.9166666665 = 2.237620264 Reorder the terms: 0.9166666665 + d = 2.237620264 Solving 0.9166666665 + d = 2.237620264 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + d = 2.237620264 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + d = 2.237620264 + -0.9166666665 d = 2.237620264 + -0.9166666665 Combine like terms: 2.237620264 + -0.9166666665 = 1.3209535975 d = 1.3209535975 Simplifying d = 1.3209535975Subproblem 2
d + 0.9166666665 = -2.237620264 Simplifying d + 0.9166666665 = -2.237620264 Reorder the terms: 0.9166666665 + d = -2.237620264 Solving 0.9166666665 + d = -2.237620264 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + d = -2.237620264 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + d = -2.237620264 + -0.9166666665 d = -2.237620264 + -0.9166666665 Combine like terms: -2.237620264 + -0.9166666665 = -3.1542869305 d = -3.1542869305 Simplifying d = -3.1542869305Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.3209535975, -3.1542869305}
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