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Simplifying 4d2 + 9d + 4 = 0 Reorder the terms: 4 + 9d + 4d2 = 0 Solving 4 + 9d + 4d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1 + 2.25d + d2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 2.25d + -1 + d2 = 0 + -1 Reorder the terms: 1 + -1 + 2.25d + d2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 2.25d + d2 = 0 + -1 2.25d + d2 = 0 + -1 Combine like terms: 0 + -1 = -1 2.25d + d2 = -1 The d term is 2.25d. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25d + 1.265625 + d2 = -1 + 1.265625 Reorder the terms: 1.265625 + 2.25d + d2 = -1 + 1.265625 Combine like terms: -1 + 1.265625 = 0.265625 1.265625 + 2.25d + d2 = 0.265625 Factor a perfect square on the left side: (d + 1.125)(d + 1.125) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (d + 1.125) equal to 0.515388203 and -0.515388203.Subproblem 1
d + 1.125 = 0.515388203 Simplifying d + 1.125 = 0.515388203 Reorder the terms: 1.125 + d = 0.515388203 Solving 1.125 + d = 0.515388203 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + d = 0.515388203 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + d = 0.515388203 + -1.125 d = 0.515388203 + -1.125 Combine like terms: 0.515388203 + -1.125 = -0.609611797 d = -0.609611797 Simplifying d = -0.609611797Subproblem 2
d + 1.125 = -0.515388203 Simplifying d + 1.125 = -0.515388203 Reorder the terms: 1.125 + d = -0.515388203 Solving 1.125 + d = -0.515388203 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + d = -0.515388203 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + d = -0.515388203 + -1.125 d = -0.515388203 + -1.125 Combine like terms: -0.515388203 + -1.125 = -1.640388203 d = -1.640388203 Simplifying d = -1.640388203Solution
The solution to the problem is based on the solutions from the subproblems. d = {-0.609611797, -1.640388203}
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