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Simplifying 6d2 + 22d = 10d + 35 Reorder the terms: 22d + 6d2 = 10d + 35 Reorder the terms: 22d + 6d2 = 35 + 10d Solving 22d + 6d2 = 35 + 10d Solving for variable 'd'. Reorder the terms: -35 + 22d + -10d + 6d2 = 35 + 10d + -35 + -10d Combine like terms: 22d + -10d = 12d -35 + 12d + 6d2 = 35 + 10d + -35 + -10d Reorder the terms: -35 + 12d + 6d2 = 35 + -35 + 10d + -10d Combine like terms: 35 + -35 = 0 -35 + 12d + 6d2 = 0 + 10d + -10d -35 + 12d + 6d2 = 10d + -10d Combine like terms: 10d + -10d = 0 -35 + 12d + 6d2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -5.833333333 + 2d + d2 = 0 Move the constant term to the right: Add '5.833333333' to each side of the equation. -5.833333333 + 2d + 5.833333333 + d2 = 0 + 5.833333333 Reorder the terms: -5.833333333 + 5.833333333 + 2d + d2 = 0 + 5.833333333 Combine like terms: -5.833333333 + 5.833333333 = 0.000000000 0.000000000 + 2d + d2 = 0 + 5.833333333 2d + d2 = 0 + 5.833333333 Combine like terms: 0 + 5.833333333 = 5.833333333 2d + d2 = 5.833333333 The d term is 2d. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2d + 1 + d2 = 5.833333333 + 1 Reorder the terms: 1 + 2d + d2 = 5.833333333 + 1 Combine like terms: 5.833333333 + 1 = 6.833333333 1 + 2d + d2 = 6.833333333 Factor a perfect square on the left side: (d + 1)(d + 1) = 6.833333333 Calculate the square root of the right side: 2.614064523 Break this problem into two subproblems by setting (d + 1) equal to 2.614064523 and -2.614064523.Subproblem 1
d + 1 = 2.614064523 Simplifying d + 1 = 2.614064523 Reorder the terms: 1 + d = 2.614064523 Solving 1 + d = 2.614064523 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + d = 2.614064523 + -1 Combine like terms: 1 + -1 = 0 0 + d = 2.614064523 + -1 d = 2.614064523 + -1 Combine like terms: 2.614064523 + -1 = 1.614064523 d = 1.614064523 Simplifying d = 1.614064523Subproblem 2
d + 1 = -2.614064523 Simplifying d + 1 = -2.614064523 Reorder the terms: 1 + d = -2.614064523 Solving 1 + d = -2.614064523 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + d = -2.614064523 + -1 Combine like terms: 1 + -1 = 0 0 + d = -2.614064523 + -1 d = -2.614064523 + -1 Combine like terms: -2.614064523 + -1 = -3.614064523 d = -3.614064523 Simplifying d = -3.614064523Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.614064523, -3.614064523}
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