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Simplifying 12y(y + 5) = 13y + 2 Reorder the terms: 12y(5 + y) = 13y + 2 (5 * 12y + y * 12y) = 13y + 2 (60y + 12y2) = 13y + 2 Reorder the terms: 60y + 12y2 = 2 + 13y Solving 60y + 12y2 = 2 + 13y Solving for variable 'y'. Reorder the terms: -2 + 60y + -13y + 12y2 = 2 + 13y + -2 + -13y Combine like terms: 60y + -13y = 47y -2 + 47y + 12y2 = 2 + 13y + -2 + -13y Reorder the terms: -2 + 47y + 12y2 = 2 + -2 + 13y + -13y Combine like terms: 2 + -2 = 0 -2 + 47y + 12y2 = 0 + 13y + -13y -2 + 47y + 12y2 = 13y + -13y Combine like terms: 13y + -13y = 0 -2 + 47y + 12y2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.1666666667 + 3.916666667y + y2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + 3.916666667y + 0.1666666667 + y2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + 3.916666667y + y2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + 3.916666667y + y2 = 0 + 0.1666666667 3.916666667y + y2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 3.916666667y + y2 = 0.1666666667 The y term is 3.916666667y. Take half its coefficient (1.958333334). Square it (3.835069447) and add it to both sides. Add '3.835069447' to each side of the equation. 3.916666667y + 3.835069447 + y2 = 0.1666666667 + 3.835069447 Reorder the terms: 3.835069447 + 3.916666667y + y2 = 0.1666666667 + 3.835069447 Combine like terms: 0.1666666667 + 3.835069447 = 4.0017361137 3.835069447 + 3.916666667y + y2 = 4.0017361137 Factor a perfect square on the left side: (y + 1.958333334)(y + 1.958333334) = 4.0017361137 Calculate the square root of the right side: 2.000433981 Break this problem into two subproblems by setting (y + 1.958333334) equal to 2.000433981 and -2.000433981.Subproblem 1
y + 1.958333334 = 2.000433981 Simplifying y + 1.958333334 = 2.000433981 Reorder the terms: 1.958333334 + y = 2.000433981 Solving 1.958333334 + y = 2.000433981 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.958333334' to each side of the equation. 1.958333334 + -1.958333334 + y = 2.000433981 + -1.958333334 Combine like terms: 1.958333334 + -1.958333334 = 0.000000000 0.000000000 + y = 2.000433981 + -1.958333334 y = 2.000433981 + -1.958333334 Combine like terms: 2.000433981 + -1.958333334 = 0.042100647 y = 0.042100647 Simplifying y = 0.042100647Subproblem 2
y + 1.958333334 = -2.000433981 Simplifying y + 1.958333334 = -2.000433981 Reorder the terms: 1.958333334 + y = -2.000433981 Solving 1.958333334 + y = -2.000433981 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.958333334' to each side of the equation. 1.958333334 + -1.958333334 + y = -2.000433981 + -1.958333334 Combine like terms: 1.958333334 + -1.958333334 = 0.000000000 0.000000000 + y = -2.000433981 + -1.958333334 y = -2.000433981 + -1.958333334 Combine like terms: -2.000433981 + -1.958333334 = -3.958767315 y = -3.958767315 Simplifying y = -3.958767315Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.042100647, -3.958767315}
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