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Simplifying -2d2 + -5d + 19 = 0 Reorder the terms: 19 + -5d + -2d2 = 0 Solving 19 + -5d + -2d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -9.5 + 2.5d + d2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + 2.5d + 9.5 + d2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + 2.5d + d2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + 2.5d + d2 = 0 + 9.5 2.5d + d2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 2.5d + d2 = 9.5 The d term is 2.5d. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5d + 1.5625 + d2 = 9.5 + 1.5625 Reorder the terms: 1.5625 + 2.5d + d2 = 9.5 + 1.5625 Combine like terms: 9.5 + 1.5625 = 11.0625 1.5625 + 2.5d + d2 = 11.0625 Factor a perfect square on the left side: (d + 1.25)(d + 1.25) = 11.0625 Calculate the square root of the right side: 3.326033674 Break this problem into two subproblems by setting (d + 1.25) equal to 3.326033674 and -3.326033674.Subproblem 1
d + 1.25 = 3.326033674 Simplifying d + 1.25 = 3.326033674 Reorder the terms: 1.25 + d = 3.326033674 Solving 1.25 + d = 3.326033674 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + d = 3.326033674 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + d = 3.326033674 + -1.25 d = 3.326033674 + -1.25 Combine like terms: 3.326033674 + -1.25 = 2.076033674 d = 2.076033674 Simplifying d = 2.076033674Subproblem 2
d + 1.25 = -3.326033674 Simplifying d + 1.25 = -3.326033674 Reorder the terms: 1.25 + d = -3.326033674 Solving 1.25 + d = -3.326033674 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + d = -3.326033674 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + d = -3.326033674 + -1.25 d = -3.326033674 + -1.25 Combine like terms: -3.326033674 + -1.25 = -4.576033674 d = -4.576033674 Simplifying d = -4.576033674Solution
The solution to the problem is based on the solutions from the subproblems. d = {2.076033674, -4.576033674}
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