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Simplifying y(y + 12) + -2(16 + -1y) = 7 + -1y(2 + y) Reorder the terms: y(12 + y) + -2(16 + -1y) = 7 + -1y(2 + y) (12 * y + y * y) + -2(16 + -1y) = 7 + -1y(2 + y) (12y + y2) + -2(16 + -1y) = 7 + -1y(2 + y) 12y + y2 + (16 * -2 + -1y * -2) = 7 + -1y(2 + y) 12y + y2 + (-32 + 2y) = 7 + -1y(2 + y) Reorder the terms: -32 + 12y + 2y + y2 = 7 + -1y(2 + y) Combine like terms: 12y + 2y = 14y -32 + 14y + y2 = 7 + -1y(2 + y) -32 + 14y + y2 = 7 + (2 * -1y + y * -1y) -32 + 14y + y2 = 7 + (-2y + -1y2) Solving -32 + 14y + y2 = 7 + -2y + -1y2 Solving for variable 'y'. Reorder the terms: -32 + -7 + 14y + 2y + y2 + y2 = 7 + -2y + -1y2 + -7 + 2y + y2 Combine like terms: -32 + -7 = -39 -39 + 14y + 2y + y2 + y2 = 7 + -2y + -1y2 + -7 + 2y + y2 Combine like terms: 14y + 2y = 16y -39 + 16y + y2 + y2 = 7 + -2y + -1y2 + -7 + 2y + y2 Combine like terms: y2 + y2 = 2y2 -39 + 16y + 2y2 = 7 + -2y + -1y2 + -7 + 2y + y2 Reorder the terms: -39 + 16y + 2y2 = 7 + -7 + -2y + 2y + -1y2 + y2 Combine like terms: 7 + -7 = 0 -39 + 16y + 2y2 = 0 + -2y + 2y + -1y2 + y2 -39 + 16y + 2y2 = -2y + 2y + -1y2 + y2 Combine like terms: -2y + 2y = 0 -39 + 16y + 2y2 = 0 + -1y2 + y2 -39 + 16y + 2y2 = -1y2 + y2 Combine like terms: -1y2 + y2 = 0 -39 + 16y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -19.5 + 8y + y2 = 0 Move the constant term to the right: Add '19.5' to each side of the equation. -19.5 + 8y + 19.5 + y2 = 0 + 19.5 Reorder the terms: -19.5 + 19.5 + 8y + y2 = 0 + 19.5 Combine like terms: -19.5 + 19.5 = 0.0 0.0 + 8y + y2 = 0 + 19.5 8y + y2 = 0 + 19.5 Combine like terms: 0 + 19.5 = 19.5 8y + y2 = 19.5 The y term is 8y. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8y + 16 + y2 = 19.5 + 16 Reorder the terms: 16 + 8y + y2 = 19.5 + 16 Combine like terms: 19.5 + 16 = 35.5 16 + 8y + y2 = 35.5 Factor a perfect square on the left side: (y + 4)(y + 4) = 35.5 Calculate the square root of the right side: 5.958187644 Break this problem into two subproblems by setting (y + 4) equal to 5.958187644 and -5.958187644.Subproblem 1
y + 4 = 5.958187644 Simplifying y + 4 = 5.958187644 Reorder the terms: 4 + y = 5.958187644 Solving 4 + y = 5.958187644 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + y = 5.958187644 + -4 Combine like terms: 4 + -4 = 0 0 + y = 5.958187644 + -4 y = 5.958187644 + -4 Combine like terms: 5.958187644 + -4 = 1.958187644 y = 1.958187644 Simplifying y = 1.958187644Subproblem 2
y + 4 = -5.958187644 Simplifying y + 4 = -5.958187644 Reorder the terms: 4 + y = -5.958187644 Solving 4 + y = -5.958187644 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + y = -5.958187644 + -4 Combine like terms: 4 + -4 = 0 0 + y = -5.958187644 + -4 y = -5.958187644 + -4 Combine like terms: -5.958187644 + -4 = -9.958187644 y = -9.958187644 Simplifying y = -9.958187644Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.958187644, -9.958187644}
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