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Simplifying -6y2 + 16y = -45 Reorder the terms: 16y + -6y2 = -45 Solving 16y + -6y2 = -45 Solving for variable 'y'. Reorder the terms: 45 + 16y + -6y2 = -45 + 45 Combine like terms: -45 + 45 = 0 45 + 16y + -6y2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -7.5 + -2.666666667y + y2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -2.666666667y + 7.5 + y2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -2.666666667y + y2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -2.666666667y + y2 = 0 + 7.5 -2.666666667y + y2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -2.666666667y + y2 = 7.5 The y term is -2.666666667y. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667y + 1.777777780 + y2 = 7.5 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667y + y2 = 7.5 + 1.777777780 Combine like terms: 7.5 + 1.777777780 = 9.27777778 1.777777780 + -2.666666667y + y2 = 9.27777778 Factor a perfect square on the left side: (y + -1.333333334)(y + -1.333333334) = 9.27777778 Calculate the square root of the right side: 3.045944481 Break this problem into two subproblems by setting (y + -1.333333334) equal to 3.045944481 and -3.045944481.Subproblem 1
y + -1.333333334 = 3.045944481 Simplifying y + -1.333333334 = 3.045944481 Reorder the terms: -1.333333334 + y = 3.045944481 Solving -1.333333334 + y = 3.045944481 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = 3.045944481 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = 3.045944481 + 1.333333334 y = 3.045944481 + 1.333333334 Combine like terms: 3.045944481 + 1.333333334 = 4.379277815 y = 4.379277815 Simplifying y = 4.379277815Subproblem 2
y + -1.333333334 = -3.045944481 Simplifying y + -1.333333334 = -3.045944481 Reorder the terms: -1.333333334 + y = -3.045944481 Solving -1.333333334 + y = -3.045944481 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = -3.045944481 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = -3.045944481 + 1.333333334 y = -3.045944481 + 1.333333334 Combine like terms: -3.045944481 + 1.333333334 = -1.712611147 y = -1.712611147 Simplifying y = -1.712611147Solution
The solution to the problem is based on the solutions from the subproblems. y = {4.379277815, -1.712611147}
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