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Simplifying 4d2 + -12d = 9 Reorder the terms: -12d + 4d2 = 9 Solving -12d + 4d2 = 9 Solving for variable 'd'. Reorder the terms: -9 + -12d + 4d2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + -12d + 4d2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.25 + -3d + d2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + -3d + 2.25 + d2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + -3d + d2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + -3d + d2 = 0 + 2.25 -3d + d2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 -3d + d2 = 2.25 The d term is -3d. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3d + 2.25 + d2 = 2.25 + 2.25 Reorder the terms: 2.25 + -3d + d2 = 2.25 + 2.25 Combine like terms: 2.25 + 2.25 = 4.5 2.25 + -3d + d2 = 4.5 Factor a perfect square on the left side: (d + -1.5)(d + -1.5) = 4.5 Calculate the square root of the right side: 2.121320344 Break this problem into two subproblems by setting (d + -1.5) equal to 2.121320344 and -2.121320344.Subproblem 1
d + -1.5 = 2.121320344 Simplifying d + -1.5 = 2.121320344 Reorder the terms: -1.5 + d = 2.121320344 Solving -1.5 + d = 2.121320344 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + d = 2.121320344 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + d = 2.121320344 + 1.5 d = 2.121320344 + 1.5 Combine like terms: 2.121320344 + 1.5 = 3.621320344 d = 3.621320344 Simplifying d = 3.621320344Subproblem 2
d + -1.5 = -2.121320344 Simplifying d + -1.5 = -2.121320344 Reorder the terms: -1.5 + d = -2.121320344 Solving -1.5 + d = -2.121320344 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + d = -2.121320344 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + d = -2.121320344 + 1.5 d = -2.121320344 + 1.5 Combine like terms: -2.121320344 + 1.5 = -0.621320344 d = -0.621320344 Simplifying d = -0.621320344Solution
The solution to the problem is based on the solutions from the subproblems. d = {3.621320344, -0.621320344}
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