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Simplifying 6 + -7d(5d + 6) = 5 Reorder the terms: 6 + -7d(6 + 5d) = 5 6 + (6 * -7d + 5d * -7d) = 5 6 + (-42d + -35d2) = 5 Solving 6 + -42d + -35d2 = 5 Solving for variable 'd'. Reorder the terms: 6 + -5 + -42d + -35d2 = 5 + -5 Combine like terms: 6 + -5 = 1 1 + -42d + -35d2 = 5 + -5 Combine like terms: 5 + -5 = 0 1 + -42d + -35d2 = 0 Begin completing the square. Divide all terms by -35 the coefficient of the squared term: Divide each side by '-35'. -0.02857142857 + 1.2d + d2 = 0 Move the constant term to the right: Add '0.02857142857' to each side of the equation. -0.02857142857 + 1.2d + 0.02857142857 + d2 = 0 + 0.02857142857 Reorder the terms: -0.02857142857 + 0.02857142857 + 1.2d + d2 = 0 + 0.02857142857 Combine like terms: -0.02857142857 + 0.02857142857 = 0.00000000000 0.00000000000 + 1.2d + d2 = 0 + 0.02857142857 1.2d + d2 = 0 + 0.02857142857 Combine like terms: 0 + 0.02857142857 = 0.02857142857 1.2d + d2 = 0.02857142857 The d term is 1.2d. Take half its coefficient (0.6). Square it (0.36) and add it to both sides. Add '0.36' to each side of the equation. 1.2d + 0.36 + d2 = 0.02857142857 + 0.36 Reorder the terms: 0.36 + 1.2d + d2 = 0.02857142857 + 0.36 Combine like terms: 0.02857142857 + 0.36 = 0.38857142857 0.36 + 1.2d + d2 = 0.38857142857 Factor a perfect square on the left side: (d + 0.6)(d + 0.6) = 0.38857142857 Calculate the square root of the right side: 0.623354978 Break this problem into two subproblems by setting (d + 0.6) equal to 0.623354978 and -0.623354978.Subproblem 1
d + 0.6 = 0.623354978 Simplifying d + 0.6 = 0.623354978 Reorder the terms: 0.6 + d = 0.623354978 Solving 0.6 + d = 0.623354978 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + d = 0.623354978 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + d = 0.623354978 + -0.6 d = 0.623354978 + -0.6 Combine like terms: 0.623354978 + -0.6 = 0.023354978 d = 0.023354978 Simplifying d = 0.023354978Subproblem 2
d + 0.6 = -0.623354978 Simplifying d + 0.6 = -0.623354978 Reorder the terms: 0.6 + d = -0.623354978 Solving 0.6 + d = -0.623354978 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + d = -0.623354978 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + d = -0.623354978 + -0.6 d = -0.623354978 + -0.6 Combine like terms: -0.623354978 + -0.6 = -1.223354978 d = -1.223354978 Simplifying d = -1.223354978Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.023354978, -1.223354978}
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