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Simplifying 13y2 + 32y + -8 = 0 Reorder the terms: -8 + 32y + 13y2 = 0 Solving -8 + 32y + 13y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -0.6153846154 + 2.461538462y + y2 = 0 Move the constant term to the right: Add '0.6153846154' to each side of the equation. -0.6153846154 + 2.461538462y + 0.6153846154 + y2 = 0 + 0.6153846154 Reorder the terms: -0.6153846154 + 0.6153846154 + 2.461538462y + y2 = 0 + 0.6153846154 Combine like terms: -0.6153846154 + 0.6153846154 = 0.0000000000 0.0000000000 + 2.461538462y + y2 = 0 + 0.6153846154 2.461538462y + y2 = 0 + 0.6153846154 Combine like terms: 0 + 0.6153846154 = 0.6153846154 2.461538462y + y2 = 0.6153846154 The y term is 2.461538462y. Take half its coefficient (1.230769231). Square it (1.514792900) and add it to both sides. Add '1.514792900' to each side of the equation. 2.461538462y + 1.514792900 + y2 = 0.6153846154 + 1.514792900 Reorder the terms: 1.514792900 + 2.461538462y + y2 = 0.6153846154 + 1.514792900 Combine like terms: 0.6153846154 + 1.514792900 = 2.1301775154 1.514792900 + 2.461538462y + y2 = 2.1301775154 Factor a perfect square on the left side: (y + 1.230769231)(y + 1.230769231) = 2.1301775154 Calculate the square root of the right side: 1.459512766 Break this problem into two subproblems by setting (y + 1.230769231) equal to 1.459512766 and -1.459512766.Subproblem 1
y + 1.230769231 = 1.459512766 Simplifying y + 1.230769231 = 1.459512766 Reorder the terms: 1.230769231 + y = 1.459512766 Solving 1.230769231 + y = 1.459512766 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.230769231' to each side of the equation. 1.230769231 + -1.230769231 + y = 1.459512766 + -1.230769231 Combine like terms: 1.230769231 + -1.230769231 = 0.000000000 0.000000000 + y = 1.459512766 + -1.230769231 y = 1.459512766 + -1.230769231 Combine like terms: 1.459512766 + -1.230769231 = 0.228743535 y = 0.228743535 Simplifying y = 0.228743535Subproblem 2
y + 1.230769231 = -1.459512766 Simplifying y + 1.230769231 = -1.459512766 Reorder the terms: 1.230769231 + y = -1.459512766 Solving 1.230769231 + y = -1.459512766 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.230769231' to each side of the equation. 1.230769231 + -1.230769231 + y = -1.459512766 + -1.230769231 Combine like terms: 1.230769231 + -1.230769231 = 0.000000000 0.000000000 + y = -1.459512766 + -1.230769231 y = -1.459512766 + -1.230769231 Combine like terms: -1.459512766 + -1.230769231 = -2.690281997 y = -2.690281997 Simplifying y = -2.690281997Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.228743535, -2.690281997}
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