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Simplifying 18d2 + 42d + 10 = 0 Reorder the terms: 10 + 42d + 18d2 = 0 Solving 10 + 42d + 18d2 = 0 Solving for variable 'd'. Factor out the Greatest Common Factor (GCF), '2'. 2(5 + 21d + 9d2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(5 + 21d + 9d2)' equal to zero and attempt to solve: Simplifying 5 + 21d + 9d2 = 0 Solving 5 + 21d + 9d2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.5555555556 + 2.333333333d + d2 = 0 Move the constant term to the right: Add '-0.5555555556' to each side of the equation. 0.5555555556 + 2.333333333d + -0.5555555556 + d2 = 0 + -0.5555555556 Reorder the terms: 0.5555555556 + -0.5555555556 + 2.333333333d + d2 = 0 + -0.5555555556 Combine like terms: 0.5555555556 + -0.5555555556 = 0.0000000000 0.0000000000 + 2.333333333d + d2 = 0 + -0.5555555556 2.333333333d + d2 = 0 + -0.5555555556 Combine like terms: 0 + -0.5555555556 = -0.5555555556 2.333333333d + d2 = -0.5555555556 The d term is 2.333333333d. Take half its coefficient (1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. 2.333333333d + 1.361111112 + d2 = -0.5555555556 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333d + d2 = -0.5555555556 + 1.361111112 Combine like terms: -0.5555555556 + 1.361111112 = 0.8055555564 1.361111112 + 2.333333333d + d2 = 0.8055555564 Factor a perfect square on the left side: (d + 1.166666667)(d + 1.166666667) = 0.8055555564 Calculate the square root of the right side: 0.897527468 Break this problem into two subproblems by setting (d + 1.166666667) equal to 0.897527468 and -0.897527468.Subproblem 1
d + 1.166666667 = 0.897527468 Simplifying d + 1.166666667 = 0.897527468 Reorder the terms: 1.166666667 + d = 0.897527468 Solving 1.166666667 + d = 0.897527468 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + d = 0.897527468 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + d = 0.897527468 + -1.166666667 d = 0.897527468 + -1.166666667 Combine like terms: 0.897527468 + -1.166666667 = -0.269139199 d = -0.269139199 Simplifying d = -0.269139199Subproblem 2
d + 1.166666667 = -0.897527468 Simplifying d + 1.166666667 = -0.897527468 Reorder the terms: 1.166666667 + d = -0.897527468 Solving 1.166666667 + d = -0.897527468 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + d = -0.897527468 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + d = -0.897527468 + -1.166666667 d = -0.897527468 + -1.166666667 Combine like terms: -0.897527468 + -1.166666667 = -2.064194135 d = -2.064194135 Simplifying d = -2.064194135Solution
The solution to the problem is based on the solutions from the subproblems. d = {-0.269139199, -2.064194135}Solution
d = {-0.269139199, -2.064194135}
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