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Simplifying (2d2 + -9d) + -1(-4d2 + 20d + 17) = 0 Reorder the terms: (-9d + 2d2) + -1(-4d2 + 20d + 17) = 0 Remove parenthesis around (-9d + 2d2) -9d + 2d2 + -1(-4d2 + 20d + 17) = 0 Reorder the terms: -9d + 2d2 + -1(17 + 20d + -4d2) = 0 -9d + 2d2 + (17 * -1 + 20d * -1 + -4d2 * -1) = 0 -9d + 2d2 + (-17 + -20d + 4d2) = 0 Reorder the terms: -17 + -9d + -20d + 2d2 + 4d2 = 0 Combine like terms: -9d + -20d = -29d -17 + -29d + 2d2 + 4d2 = 0 Combine like terms: 2d2 + 4d2 = 6d2 -17 + -29d + 6d2 = 0 Solving -17 + -29d + 6d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.833333333 + -4.833333333d + d2 = 0 Move the constant term to the right: Add '2.833333333' to each side of the equation. -2.833333333 + -4.833333333d + 2.833333333 + d2 = 0 + 2.833333333 Reorder the terms: -2.833333333 + 2.833333333 + -4.833333333d + d2 = 0 + 2.833333333 Combine like terms: -2.833333333 + 2.833333333 = 0.000000000 0.000000000 + -4.833333333d + d2 = 0 + 2.833333333 -4.833333333d + d2 = 0 + 2.833333333 Combine like terms: 0 + 2.833333333 = 2.833333333 -4.833333333d + d2 = 2.833333333 The d term is -4.833333333d. Take half its coefficient (-2.416666667). Square it (5.840277779) and add it to both sides. Add '5.840277779' to each side of the equation. -4.833333333d + 5.840277779 + d2 = 2.833333333 + 5.840277779 Reorder the terms: 5.840277779 + -4.833333333d + d2 = 2.833333333 + 5.840277779 Combine like terms: 2.833333333 + 5.840277779 = 8.673611112 5.840277779 + -4.833333333d + d2 = 8.673611112 Factor a perfect square on the left side: (d + -2.416666667)(d + -2.416666667) = 8.673611112 Calculate the square root of the right side: 2.945099508 Break this problem into two subproblems by setting (d + -2.416666667) equal to 2.945099508 and -2.945099508.Subproblem 1
d + -2.416666667 = 2.945099508 Simplifying d + -2.416666667 = 2.945099508 Reorder the terms: -2.416666667 + d = 2.945099508 Solving -2.416666667 + d = 2.945099508 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '2.416666667' to each side of the equation. -2.416666667 + 2.416666667 + d = 2.945099508 + 2.416666667 Combine like terms: -2.416666667 + 2.416666667 = 0.000000000 0.000000000 + d = 2.945099508 + 2.416666667 d = 2.945099508 + 2.416666667 Combine like terms: 2.945099508 + 2.416666667 = 5.361766175 d = 5.361766175 Simplifying d = 5.361766175Subproblem 2
d + -2.416666667 = -2.945099508 Simplifying d + -2.416666667 = -2.945099508 Reorder the terms: -2.416666667 + d = -2.945099508 Solving -2.416666667 + d = -2.945099508 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '2.416666667' to each side of the equation. -2.416666667 + 2.416666667 + d = -2.945099508 + 2.416666667 Combine like terms: -2.416666667 + 2.416666667 = 0.000000000 0.000000000 + d = -2.945099508 + 2.416666667 d = -2.945099508 + 2.416666667 Combine like terms: -2.945099508 + 2.416666667 = -0.528432841 d = -0.528432841 Simplifying d = -0.528432841Solution
The solution to the problem is based on the solutions from the subproblems. d = {5.361766175, -0.528432841}
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