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Simplifying -4d(d + 4) = -32 Reorder the terms: -4d(4 + d) = -32 (4 * -4d + d * -4d) = -32 (-16d + -4d2) = -32 Solving -16d + -4d2 = -32 Solving for variable 'd'. Reorder the terms: 32 + -16d + -4d2 = -32 + 32 Combine like terms: -32 + 32 = 0 32 + -16d + -4d2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(8 + -4d + -1d2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(8 + -4d + -1d2)' equal to zero and attempt to solve: Simplifying 8 + -4d + -1d2 = 0 Solving 8 + -4d + -1d2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -8 + 4d + d2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 4d + 8 + d2 = 0 + 8 Reorder the terms: -8 + 8 + 4d + d2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 4d + d2 = 0 + 8 4d + d2 = 0 + 8 Combine like terms: 0 + 8 = 8 4d + d2 = 8 The d term is 4d. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4d + 4 + d2 = 8 + 4 Reorder the terms: 4 + 4d + d2 = 8 + 4 Combine like terms: 8 + 4 = 12 4 + 4d + d2 = 12 Factor a perfect square on the left side: (d + 2)(d + 2) = 12 Calculate the square root of the right side: 3.464101615 Break this problem into two subproblems by setting (d + 2) equal to 3.464101615 and -3.464101615.Subproblem 1
d + 2 = 3.464101615 Simplifying d + 2 = 3.464101615 Reorder the terms: 2 + d = 3.464101615 Solving 2 + d = 3.464101615 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + d = 3.464101615 + -2 Combine like terms: 2 + -2 = 0 0 + d = 3.464101615 + -2 d = 3.464101615 + -2 Combine like terms: 3.464101615 + -2 = 1.464101615 d = 1.464101615 Simplifying d = 1.464101615Subproblem 2
d + 2 = -3.464101615 Simplifying d + 2 = -3.464101615 Reorder the terms: 2 + d = -3.464101615 Solving 2 + d = -3.464101615 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + d = -3.464101615 + -2 Combine like terms: 2 + -2 = 0 0 + d = -3.464101615 + -2 d = -3.464101615 + -2 Combine like terms: -3.464101615 + -2 = -5.464101615 d = -5.464101615 Simplifying d = -5.464101615Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.464101615, -5.464101615}Solution
d = {1.464101615, -5.464101615}
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