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Simplifying 27 + -1(36 + -1y) * y = 12 Reorder the terms for easier multiplication: 27 + -1y(36 + -1y) = 12 27 + (36 * -1y + -1y * -1y) = 12 27 + (-36y + 1y2) = 12 Solving 27 + -36y + 1y2 = 12 Solving for variable 'y'. Reorder the terms: 27 + -12 + -36y + 1y2 = 12 + -12 Combine like terms: 27 + -12 = 15 15 + -36y + 1y2 = 12 + -12 Combine like terms: 12 + -12 = 0 15 + -36y + 1y2 = 0 Begin completing the square. Move the constant term to the right: Add '-15' to each side of the equation. 15 + -36y + -15 + y2 = 0 + -15 Reorder the terms: 15 + -15 + -36y + y2 = 0 + -15 Combine like terms: 15 + -15 = 0 0 + -36y + y2 = 0 + -15 -36y + y2 = 0 + -15 Combine like terms: 0 + -15 = -15 -36y + y2 = -15 The y term is -36y. Take half its coefficient (-18). Square it (324) and add it to both sides. Add '324' to each side of the equation. -36y + 324 + y2 = -15 + 324 Reorder the terms: 324 + -36y + y2 = -15 + 324 Combine like terms: -15 + 324 = 309 324 + -36y + y2 = 309 Factor a perfect square on the left side: (y + -18)(y + -18) = 309 Calculate the square root of the right side: 17.578395831 Break this problem into two subproblems by setting (y + -18) equal to 17.578395831 and -17.578395831.Subproblem 1
y + -18 = 17.578395831 Simplifying y + -18 = 17.578395831 Reorder the terms: -18 + y = 17.578395831 Solving -18 + y = 17.578395831 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '18' to each side of the equation. -18 + 18 + y = 17.578395831 + 18 Combine like terms: -18 + 18 = 0 0 + y = 17.578395831 + 18 y = 17.578395831 + 18 Combine like terms: 17.578395831 + 18 = 35.578395831 y = 35.578395831 Simplifying y = 35.578395831Subproblem 2
y + -18 = -17.578395831 Simplifying y + -18 = -17.578395831 Reorder the terms: -18 + y = -17.578395831 Solving -18 + y = -17.578395831 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '18' to each side of the equation. -18 + 18 + y = -17.578395831 + 18 Combine like terms: -18 + 18 = 0 0 + y = -17.578395831 + 18 y = -17.578395831 + 18 Combine like terms: -17.578395831 + 18 = 0.421604169 y = 0.421604169 Simplifying y = 0.421604169Solution
The solution to the problem is based on the solutions from the subproblems. y = {35.578395831, 0.421604169}
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