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Simplifying -7d2 + 7d + 8 = 0 Reorder the terms: 8 + 7d + -7d2 = 0 Solving 8 + 7d + -7d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -1.142857143 + -1d + d2 = 0 Move the constant term to the right: Add '1.142857143' to each side of the equation. -1.142857143 + -1d + 1.142857143 + d2 = 0 + 1.142857143 Reorder the terms: -1.142857143 + 1.142857143 + -1d + d2 = 0 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + -1d + d2 = 0 + 1.142857143 -1d + d2 = 0 + 1.142857143 Combine like terms: 0 + 1.142857143 = 1.142857143 -1d + d2 = 1.142857143 The d term is -1d. 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. -1d + 0.25 + d2 = 1.142857143 + 0.25 Reorder the terms: 0.25 + -1d + d2 = 1.142857143 + 0.25 Combine like terms: 1.142857143 + 0.25 = 1.392857143 0.25 + -1d + d2 = 1.392857143 Factor a perfect square on the left side: (d + -0.5)(d + -0.5) = 1.392857143 Calculate the square root of the right side: 1.180193689 Break this problem into two subproblems by setting (d + -0.5) equal to 1.180193689 and -1.180193689.Subproblem 1
d + -0.5 = 1.180193689 Simplifying d + -0.5 = 1.180193689 Reorder the terms: -0.5 + d = 1.180193689 Solving -0.5 + d = 1.180193689 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 = 1.180193689 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = 1.180193689 + 0.5 d = 1.180193689 + 0.5 Combine like terms: 1.180193689 + 0.5 = 1.680193689 d = 1.680193689 Simplifying d = 1.680193689Subproblem 2
d + -0.5 = -1.180193689 Simplifying d + -0.5 = -1.180193689 Reorder the terms: -0.5 + d = -1.180193689 Solving -0.5 + d = -1.180193689 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 = -1.180193689 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = -1.180193689 + 0.5 d = -1.180193689 + 0.5 Combine like terms: -1.180193689 + 0.5 = -0.680193689 d = -0.680193689 Simplifying d = -0.680193689Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.680193689, -0.680193689}
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