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Simplifying 5d2 + 5d + 1 = 0 Reorder the terms: 1 + 5d + 5d2 = 0 Solving 1 + 5d + 5d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + d + d2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + d + -0.2 + d2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + d + d2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + d + d2 = 0 + -0.2 d + d2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 d + d2 = -0.2 The d term is d. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + d2 = -0.2 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + d2 = -0.2 + 0.25 Combine like terms: -0.2 + 0.25 = 0.05 1.25 + d2 = 0.05 Factor a perfect square on the left side: (d + 0.5)(d + 0.5) = 0.05 Calculate the square root of the right side: 0.223606798 Break this problem into two subproblems by setting (d + 0.5) equal to 0.223606798 and -0.223606798.Subproblem 1
d + 0.5 = 0.223606798 Simplifying d + 0.5 = 0.223606798 Reorder the terms: 0.5 + d = 0.223606798 Solving 0.5 + d = 0.223606798 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + d = 0.223606798 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + d = 0.223606798 + -0.5 d = 0.223606798 + -0.5 Combine like terms: 0.223606798 + -0.5 = -0.276393202 d = -0.276393202 Simplifying d = -0.276393202Subproblem 2
d + 0.5 = -0.223606798 Simplifying d + 0.5 = -0.223606798 Reorder the terms: 0.5 + d = -0.223606798 Solving 0.5 + d = -0.223606798 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + d = -0.223606798 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + d = -0.223606798 + -0.5 d = -0.223606798 + -0.5 Combine like terms: -0.223606798 + -0.5 = -0.723606798 d = -0.723606798 Simplifying d = -0.723606798Solution
The solution to the problem is based on the solutions from the subproblems. d = {-0.276393202, -0.723606798}
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