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Simplifying 0.625d + 0.6666666 = 0.75d(d + -2) Reorder the terms: 0.6666666 + 0.625d = 0.75d(d + -2) Reorder the terms: 0.6666666 + 0.625d = 0.75d(-2 + d) 0.6666666 + 0.625d = (-2 * 0.75d + d * 0.75d) 0.6666666 + 0.625d = (-1.5d + 0.75d2) Solving 0.6666666 + 0.625d = -1.5d + 0.75d2 Solving for variable 'd'. Combine like terms: 0.625d + 1.5d = 2.125d 0.6666666 + 2.125d + -0.75d2 = -1.5d + 0.75d2 + 1.5d + -0.75d2 Reorder the terms: 0.6666666 + 2.125d + -0.75d2 = -1.5d + 1.5d + 0.75d2 + -0.75d2 Combine like terms: -1.5d + 1.5d = 0.0 0.6666666 + 2.125d + -0.75d2 = 0.0 + 0.75d2 + -0.75d2 0.6666666 + 2.125d + -0.75d2 = 0.75d2 + -0.75d2 Combine like terms: 0.75d2 + -0.75d2 = 0.00 0.6666666 + 2.125d + -0.75d2 = 0.00 Begin completing the square. Divide all terms by -0.75 the coefficient of the squared term: Divide each side by '-0.75'. -0.8888888 + -2.833333333d + d2 = 0 Move the constant term to the right: Add '0.8888888' to each side of the equation. -0.8888888 + -2.833333333d + 0.8888888 + d2 = 0 + 0.8888888 Reorder the terms: -0.8888888 + 0.8888888 + -2.833333333d + d2 = 0 + 0.8888888 Combine like terms: -0.8888888 + 0.8888888 = 0.0000000 0.0000000 + -2.833333333d + d2 = 0 + 0.8888888 -2.833333333d + d2 = 0 + 0.8888888 Combine like terms: 0 + 0.8888888 = 0.8888888 -2.833333333d + d2 = 0.8888888 The d term is -2.833333333d. Take half its coefficient (-1.416666667). Square it (2.006944445) and add it to both sides. Add '2.006944445' to each side of the equation. -2.833333333d + 2.006944445 + d2 = 0.8888888 + 2.006944445 Reorder the terms: 2.006944445 + -2.833333333d + d2 = 0.8888888 + 2.006944445 Combine like terms: 0.8888888 + 2.006944445 = 2.895833245 2.006944445 + -2.833333333d + d2 = 2.895833245 Factor a perfect square on the left side: (d + -1.416666667)(d + -1.416666667) = 2.895833245 Calculate the square root of the right side: 1.701714795 Break this problem into two subproblems by setting (d + -1.416666667) equal to 1.701714795 and -1.701714795.Subproblem 1
d + -1.416666667 = 1.701714795 Simplifying d + -1.416666667 = 1.701714795 Reorder the terms: -1.416666667 + d = 1.701714795 Solving -1.416666667 + d = 1.701714795 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '1.416666667' to each side of the equation. -1.416666667 + 1.416666667 + d = 1.701714795 + 1.416666667 Combine like terms: -1.416666667 + 1.416666667 = 0.000000000 0.000000000 + d = 1.701714795 + 1.416666667 d = 1.701714795 + 1.416666667 Combine like terms: 1.701714795 + 1.416666667 = 3.118381462 d = 3.118381462 Simplifying d = 3.118381462Subproblem 2
d + -1.416666667 = -1.701714795 Simplifying d + -1.416666667 = -1.701714795 Reorder the terms: -1.416666667 + d = -1.701714795 Solving -1.416666667 + d = -1.701714795 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '1.416666667' to each side of the equation. -1.416666667 + 1.416666667 + d = -1.701714795 + 1.416666667 Combine like terms: -1.416666667 + 1.416666667 = 0.000000000 0.000000000 + d = -1.701714795 + 1.416666667 d = -1.701714795 + 1.416666667 Combine like terms: -1.701714795 + 1.416666667 = -0.285048128 d = -0.285048128 Simplifying d = -0.285048128Solution
The solution to the problem is based on the solutions from the subproblems. d = {3.118381462, -0.285048128}
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