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Simplifying 5d2 + 9d + 3 = 0 Reorder the terms: 3 + 9d + 5d2 = 0 Solving 3 + 9d + 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'. 0.6 + 1.8d + d2 = 0 Move the constant term to the right: Add '-0.6' to each side of the equation. 0.6 + 1.8d + -0.6 + d2 = 0 + -0.6 Reorder the terms: 0.6 + -0.6 + 1.8d + d2 = 0 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + 1.8d + d2 = 0 + -0.6 1.8d + d2 = 0 + -0.6 Combine like terms: 0 + -0.6 = -0.6 1.8d + d2 = -0.6 The d term is 1.8d. Take half its coefficient (0.9). Square it (0.81) and add it to both sides. Add '0.81' to each side of the equation. 1.8d + 0.81 + d2 = -0.6 + 0.81 Reorder the terms: 0.81 + 1.8d + d2 = -0.6 + 0.81 Combine like terms: -0.6 + 0.81 = 0.21 0.81 + 1.8d + d2 = 0.21 Factor a perfect square on the left side: (d + 0.9)(d + 0.9) = 0.21 Calculate the square root of the right side: 0.458257569 Break this problem into two subproblems by setting (d + 0.9) equal to 0.458257569 and -0.458257569.Subproblem 1
d + 0.9 = 0.458257569 Simplifying d + 0.9 = 0.458257569 Reorder the terms: 0.9 + d = 0.458257569 Solving 0.9 + d = 0.458257569 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + d = 0.458257569 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + d = 0.458257569 + -0.9 d = 0.458257569 + -0.9 Combine like terms: 0.458257569 + -0.9 = -0.441742431 d = -0.441742431 Simplifying d = -0.441742431Subproblem 2
d + 0.9 = -0.458257569 Simplifying d + 0.9 = -0.458257569 Reorder the terms: 0.9 + d = -0.458257569 Solving 0.9 + d = -0.458257569 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.9' to each side of the equation. 0.9 + -0.9 + d = -0.458257569 + -0.9 Combine like terms: 0.9 + -0.9 = 0.0 0.0 + d = -0.458257569 + -0.9 d = -0.458257569 + -0.9 Combine like terms: -0.458257569 + -0.9 = -1.358257569 d = -1.358257569 Simplifying d = -1.358257569Solution
The solution to the problem is based on the solutions from the subproblems. d = {-0.441742431, -1.358257569}
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