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Simplifying 6 + -4d2 + -19 + 3d + 9d2 = 0 Reorder the terms: 6 + -19 + 3d + -4d2 + 9d2 = 0 Combine like terms: 6 + -19 = -13 -13 + 3d + -4d2 + 9d2 = 0 Combine like terms: -4d2 + 9d2 = 5d2 -13 + 3d + 5d2 = 0 Solving -13 + 3d + 5d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.6 + 0.6d + d2 = 0 Move the constant term to the right: Add '2.6' to each side of the equation. -2.6 + 0.6d + 2.6 + d2 = 0 + 2.6 Reorder the terms: -2.6 + 2.6 + 0.6d + d2 = 0 + 2.6 Combine like terms: -2.6 + 2.6 = 0.0 0.0 + 0.6d + d2 = 0 + 2.6 0.6d + d2 = 0 + 2.6 Combine like terms: 0 + 2.6 = 2.6 0.6d + d2 = 2.6 The d term is 0.6d. Take half its coefficient (0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. 0.6d + 0.09 + d2 = 2.6 + 0.09 Reorder the terms: 0.09 + 0.6d + d2 = 2.6 + 0.09 Combine like terms: 2.6 + 0.09 = 2.69 0.09 + 0.6d + d2 = 2.69 Factor a perfect square on the left side: (d + 0.3)(d + 0.3) = 2.69 Calculate the square root of the right side: 1.640121947 Break this problem into two subproblems by setting (d + 0.3) equal to 1.640121947 and -1.640121947.Subproblem 1
d + 0.3 = 1.640121947 Simplifying d + 0.3 = 1.640121947 Reorder the terms: 0.3 + d = 1.640121947 Solving 0.3 + d = 1.640121947 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + d = 1.640121947 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + d = 1.640121947 + -0.3 d = 1.640121947 + -0.3 Combine like terms: 1.640121947 + -0.3 = 1.340121947 d = 1.340121947 Simplifying d = 1.340121947Subproblem 2
d + 0.3 = -1.640121947 Simplifying d + 0.3 = -1.640121947 Reorder the terms: 0.3 + d = -1.640121947 Solving 0.3 + d = -1.640121947 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + d = -1.640121947 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + d = -1.640121947 + -0.3 d = -1.640121947 + -0.3 Combine like terms: -1.640121947 + -0.3 = -1.940121947 d = -1.940121947 Simplifying d = -1.940121947Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.340121947, -1.940121947}
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