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Simplifying (12y2 + 19y + -14) + -1(18y2 + -3y + -17) = 0 Reorder the terms: (-14 + 19y + 12y2) + -1(18y2 + -3y + -17) = 0 Remove parenthesis around (-14 + 19y + 12y2) -14 + 19y + 12y2 + -1(18y2 + -3y + -17) = 0 Reorder the terms: -14 + 19y + 12y2 + -1(-17 + -3y + 18y2) = 0 -14 + 19y + 12y2 + (-17 * -1 + -3y * -1 + 18y2 * -1) = 0 -14 + 19y + 12y2 + (17 + 3y + -18y2) = 0 Reorder the terms: -14 + 17 + 19y + 3y + 12y2 + -18y2 = 0 Combine like terms: -14 + 17 = 3 3 + 19y + 3y + 12y2 + -18y2 = 0 Combine like terms: 19y + 3y = 22y 3 + 22y + 12y2 + -18y2 = 0 Combine like terms: 12y2 + -18y2 = -6y2 3 + 22y + -6y2 = 0 Solving 3 + 22y + -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'. -0.5 + -3.666666667y + y2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -3.666666667y + 0.5 + y2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -3.666666667y + y2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -3.666666667y + y2 = 0 + 0.5 -3.666666667y + y2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -3.666666667y + y2 = 0.5 The y term is -3.666666667y. Take half its coefficient (-1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. -3.666666667y + 3.361111114 + y2 = 0.5 + 3.361111114 Reorder the terms: 3.361111114 + -3.666666667y + y2 = 0.5 + 3.361111114 Combine like terms: 0.5 + 3.361111114 = 3.861111114 3.361111114 + -3.666666667y + y2 = 3.861111114 Factor a perfect square on the left side: (y + -1.833333334)(y + -1.833333334) = 3.861111114 Calculate the square root of the right side: 1.964971021 Break this problem into two subproblems by setting (y + -1.833333334) equal to 1.964971021 and -1.964971021.Subproblem 1
y + -1.833333334 = 1.964971021 Simplifying y + -1.833333334 = 1.964971021 Reorder the terms: -1.833333334 + y = 1.964971021 Solving -1.833333334 + y = 1.964971021 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + y = 1.964971021 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + y = 1.964971021 + 1.833333334 y = 1.964971021 + 1.833333334 Combine like terms: 1.964971021 + 1.833333334 = 3.798304355 y = 3.798304355 Simplifying y = 3.798304355Subproblem 2
y + -1.833333334 = -1.964971021 Simplifying y + -1.833333334 = -1.964971021 Reorder the terms: -1.833333334 + y = -1.964971021 Solving -1.833333334 + y = -1.964971021 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + y = -1.964971021 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + y = -1.964971021 + 1.833333334 y = -1.964971021 + 1.833333334 Combine like terms: -1.964971021 + 1.833333334 = -0.131637687 y = -0.131637687 Simplifying y = -0.131637687Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.798304355, -0.131637687}
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