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Simplifying 5(d + 4) = 3(d + 5) * 2d Reorder the terms: 5(4 + d) = 3(d + 5) * 2d (4 * 5 + d * 5) = 3(d + 5) * 2d (20 + 5d) = 3(d + 5) * 2d Reorder the terms: 20 + 5d = 3(5 + d) * 2d Reorder the terms for easier multiplication: 20 + 5d = 3 * 2d(5 + d) Multiply 3 * 2 20 + 5d = 6d(5 + d) 20 + 5d = (5 * 6d + d * 6d) 20 + 5d = (30d + 6d2) Solving 20 + 5d = 30d + 6d2 Solving for variable 'd'. Combine like terms: 5d + -30d = -25d 20 + -25d + -6d2 = 30d + 6d2 + -30d + -6d2 Reorder the terms: 20 + -25d + -6d2 = 30d + -30d + 6d2 + -6d2 Combine like terms: 30d + -30d = 0 20 + -25d + -6d2 = 0 + 6d2 + -6d2 20 + -25d + -6d2 = 6d2 + -6d2 Combine like terms: 6d2 + -6d2 = 0 20 + -25d + -6d2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -3.333333333 + 4.166666667d + d2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + 4.166666667d + 3.333333333 + d2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + 4.166666667d + d2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + 4.166666667d + d2 = 0 + 3.333333333 4.166666667d + d2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 4.166666667d + d2 = 3.333333333 The d term is 4.166666667d. Take half its coefficient (2.083333334). Square it (4.340277781) and add it to both sides. Add '4.340277781' to each side of the equation. 4.166666667d + 4.340277781 + d2 = 3.333333333 + 4.340277781 Reorder the terms: 4.340277781 + 4.166666667d + d2 = 3.333333333 + 4.340277781 Combine like terms: 3.333333333 + 4.340277781 = 7.673611114 4.340277781 + 4.166666667d + d2 = 7.673611114 Factor a perfect square on the left side: (d + 2.083333334)(d + 2.083333334) = 7.673611114 Calculate the square root of the right side: 2.770128357 Break this problem into two subproblems by setting (d + 2.083333334) equal to 2.770128357 and -2.770128357.Subproblem 1
d + 2.083333334 = 2.770128357 Simplifying d + 2.083333334 = 2.770128357 Reorder the terms: 2.083333334 + d = 2.770128357 Solving 2.083333334 + d = 2.770128357 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.083333334' to each side of the equation. 2.083333334 + -2.083333334 + d = 2.770128357 + -2.083333334 Combine like terms: 2.083333334 + -2.083333334 = 0.000000000 0.000000000 + d = 2.770128357 + -2.083333334 d = 2.770128357 + -2.083333334 Combine like terms: 2.770128357 + -2.083333334 = 0.686795023 d = 0.686795023 Simplifying d = 0.686795023Subproblem 2
d + 2.083333334 = -2.770128357 Simplifying d + 2.083333334 = -2.770128357 Reorder the terms: 2.083333334 + d = -2.770128357 Solving 2.083333334 + d = -2.770128357 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.083333334' to each side of the equation. 2.083333334 + -2.083333334 + d = -2.770128357 + -2.083333334 Combine like terms: 2.083333334 + -2.083333334 = 0.000000000 0.000000000 + d = -2.770128357 + -2.083333334 d = -2.770128357 + -2.083333334 Combine like terms: -2.770128357 + -2.083333334 = -4.853461691 d = -4.853461691 Simplifying d = -4.853461691Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.686795023, -4.853461691}
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