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Simplifying 6y2 + -23y + -145 = 0 Reorder the terms: -145 + -23y + 6y2 = 0 Solving -145 + -23y + 6y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -24.16666667 + -3.833333333y + y2 = 0 Move the constant term to the right: Add '24.16666667' to each side of the equation. -24.16666667 + -3.833333333y + 24.16666667 + y2 = 0 + 24.16666667 Reorder the terms: -24.16666667 + 24.16666667 + -3.833333333y + y2 = 0 + 24.16666667 Combine like terms: -24.16666667 + 24.16666667 = 0.00000000 0.00000000 + -3.833333333y + y2 = 0 + 24.16666667 -3.833333333y + y2 = 0 + 24.16666667 Combine like terms: 0 + 24.16666667 = 24.16666667 -3.833333333y + y2 = 24.16666667 The y term is -3.833333333y. Take half its coefficient (-1.916666667). Square it (3.673611112) and add it to both sides. Add '3.673611112' to each side of the equation. -3.833333333y + 3.673611112 + y2 = 24.16666667 + 3.673611112 Reorder the terms: 3.673611112 + -3.833333333y + y2 = 24.16666667 + 3.673611112 Combine like terms: 24.16666667 + 3.673611112 = 27.840277782 3.673611112 + -3.833333333y + y2 = 27.840277782 Factor a perfect square on the left side: (y + -1.916666667)(y + -1.916666667) = 27.840277782 Calculate the square root of the right side: 5.276388706 Break this problem into two subproblems by setting (y + -1.916666667) equal to 5.276388706 and -5.276388706.Subproblem 1
y + -1.916666667 = 5.276388706 Simplifying y + -1.916666667 = 5.276388706 Reorder the terms: -1.916666667 + y = 5.276388706 Solving -1.916666667 + y = 5.276388706 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.916666667' to each side of the equation. -1.916666667 + 1.916666667 + y = 5.276388706 + 1.916666667 Combine like terms: -1.916666667 + 1.916666667 = 0.000000000 0.000000000 + y = 5.276388706 + 1.916666667 y = 5.276388706 + 1.916666667 Combine like terms: 5.276388706 + 1.916666667 = 7.193055373 y = 7.193055373 Simplifying y = 7.193055373Subproblem 2
y + -1.916666667 = -5.276388706 Simplifying y + -1.916666667 = -5.276388706 Reorder the terms: -1.916666667 + y = -5.276388706 Solving -1.916666667 + y = -5.276388706 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.916666667' to each side of the equation. -1.916666667 + 1.916666667 + y = -5.276388706 + 1.916666667 Combine like terms: -1.916666667 + 1.916666667 = 0.000000000 0.000000000 + y = -5.276388706 + 1.916666667 y = -5.276388706 + 1.916666667 Combine like terms: -5.276388706 + 1.916666667 = -3.359722039 y = -3.359722039 Simplifying y = -3.359722039Solution
The solution to the problem is based on the solutions from the subproblems. y = {7.193055373, -3.359722039}
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