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Simplifying (d2 + -9) + -1(-5d2 + 3d + -1) = 0 Reorder the terms: (-9 + d2) + -1(-5d2 + 3d + -1) = 0 Remove parenthesis around (-9 + d2) -9 + d2 + -1(-5d2 + 3d + -1) = 0 Reorder the terms: -9 + d2 + -1(-1 + 3d + -5d2) = 0 -9 + d2 + (-1 * -1 + 3d * -1 + -5d2 * -1) = 0 -9 + d2 + (1 + -3d + 5d2) = 0 Reorder the terms: -9 + 1 + -3d + d2 + 5d2 = 0 Combine like terms: -9 + 1 = -8 -8 + -3d + d2 + 5d2 = 0 Combine like terms: d2 + 5d2 = 6d2 -8 + -3d + 6d2 = 0 Solving -8 + -3d + 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'. -1.333333333 + -0.5d + d2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -0.5d + 1.333333333 + d2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -0.5d + d2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -0.5d + d2 = 0 + 1.333333333 -0.5d + d2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -0.5d + d2 = 1.333333333 The d term is -0.5d. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5d + 0.0625 + d2 = 1.333333333 + 0.0625 Reorder the terms: 0.0625 + -0.5d + d2 = 1.333333333 + 0.0625 Combine like terms: 1.333333333 + 0.0625 = 1.395833333 0.0625 + -0.5d + d2 = 1.395833333 Factor a perfect square on the left side: (d + -0.25)(d + -0.25) = 1.395833333 Calculate the square root of the right side: 1.181453906 Break this problem into two subproblems by setting (d + -0.25) equal to 1.181453906 and -1.181453906.Subproblem 1
d + -0.25 = 1.181453906 Simplifying d + -0.25 = 1.181453906 Reorder the terms: -0.25 + d = 1.181453906 Solving -0.25 + d = 1.181453906 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + d = 1.181453906 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + d = 1.181453906 + 0.25 d = 1.181453906 + 0.25 Combine like terms: 1.181453906 + 0.25 = 1.431453906 d = 1.431453906 Simplifying d = 1.431453906Subproblem 2
d + -0.25 = -1.181453906 Simplifying d + -0.25 = -1.181453906 Reorder the terms: -0.25 + d = -1.181453906 Solving -0.25 + d = -1.181453906 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + d = -1.181453906 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + d = -1.181453906 + 0.25 d = -1.181453906 + 0.25 Combine like terms: -1.181453906 + 0.25 = -0.931453906 d = -0.931453906 Simplifying d = -0.931453906Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.431453906, -0.931453906}
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