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Simplifying -5d(d + 3) = 7d + 5 Reorder the terms: -5d(3 + d) = 7d + 5 (3 * -5d + d * -5d) = 7d + 5 (-15d + -5d2) = 7d + 5 Reorder the terms: -15d + -5d2 = 5 + 7d Solving -15d + -5d2 = 5 + 7d Solving for variable 'd'. Reorder the terms: -5 + -15d + -7d + -5d2 = 5 + 7d + -5 + -7d Combine like terms: -15d + -7d = -22d -5 + -22d + -5d2 = 5 + 7d + -5 + -7d Reorder the terms: -5 + -22d + -5d2 = 5 + -5 + 7d + -7d Combine like terms: 5 + -5 = 0 -5 + -22d + -5d2 = 0 + 7d + -7d -5 + -22d + -5d2 = 7d + -7d Combine like terms: 7d + -7d = 0 -5 + -22d + -5d2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(5 + 22d + 5d2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(5 + 22d + 5d2)' equal to zero and attempt to solve: Simplifying 5 + 22d + 5d2 = 0 Solving 5 + 22d + 5d2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 1 + 4.4d + d2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 4.4d + -1 + d2 = 0 + -1 Reorder the terms: 1 + -1 + 4.4d + d2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 4.4d + d2 = 0 + -1 4.4d + d2 = 0 + -1 Combine like terms: 0 + -1 = -1 4.4d + d2 = -1 The d term is 4.4d. Take half its coefficient (2.2). Square it (4.84) and add it to both sides. Add '4.84' to each side of the equation. 4.4d + 4.84 + d2 = -1 + 4.84 Reorder the terms: 4.84 + 4.4d + d2 = -1 + 4.84 Combine like terms: -1 + 4.84 = 3.84 4.84 + 4.4d + d2 = 3.84 Factor a perfect square on the left side: (d + 2.2)(d + 2.2) = 3.84 Calculate the square root of the right side: 1.959591794 Break this problem into two subproblems by setting (d + 2.2) equal to 1.959591794 and -1.959591794.Subproblem 1
d + 2.2 = 1.959591794 Simplifying d + 2.2 = 1.959591794 Reorder the terms: 2.2 + d = 1.959591794 Solving 2.2 + d = 1.959591794 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.2' to each side of the equation. 2.2 + -2.2 + d = 1.959591794 + -2.2 Combine like terms: 2.2 + -2.2 = 0.0 0.0 + d = 1.959591794 + -2.2 d = 1.959591794 + -2.2 Combine like terms: 1.959591794 + -2.2 = -0.240408206 d = -0.240408206 Simplifying d = -0.240408206Subproblem 2
d + 2.2 = -1.959591794 Simplifying d + 2.2 = -1.959591794 Reorder the terms: 2.2 + d = -1.959591794 Solving 2.2 + d = -1.959591794 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.2' to each side of the equation. 2.2 + -2.2 + d = -1.959591794 + -2.2 Combine like terms: 2.2 + -2.2 = 0.0 0.0 + d = -1.959591794 + -2.2 d = -1.959591794 + -2.2 Combine like terms: -1.959591794 + -2.2 = -4.159591794 d = -4.159591794 Simplifying d = -4.159591794Solution
The solution to the problem is based on the solutions from the subproblems. d = {-0.240408206, -4.159591794}Solution
d = {-0.240408206, -4.159591794}
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