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Simplifying (2y + 7) * 2y = -3 Reorder the terms: (7 + 2y) * 2y = -3 Reorder the terms for easier multiplication: 2y(7 + 2y) = -3 (7 * 2y + 2y * 2y) = -3 (14y + 4y2) = -3 Solving 14y + 4y2 = -3 Solving for variable 'y'. Reorder the terms: 3 + 14y + 4y2 = -3 + 3 Combine like terms: -3 + 3 = 0 3 + 14y + 4y2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + 3.5y + y2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 3.5y + -0.75 + y2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 3.5y + y2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 3.5y + y2 = 0 + -0.75 3.5y + y2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 3.5y + y2 = -0.75 The y term is 3.5y. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5y + 3.0625 + y2 = -0.75 + 3.0625 Reorder the terms: 3.0625 + 3.5y + y2 = -0.75 + 3.0625 Combine like terms: -0.75 + 3.0625 = 2.3125 3.0625 + 3.5y + y2 = 2.3125 Factor a perfect square on the left side: (y + 1.75)(y + 1.75) = 2.3125 Calculate the square root of the right side: 1.520690633 Break this problem into two subproblems by setting (y + 1.75) equal to 1.520690633 and -1.520690633.Subproblem 1
y + 1.75 = 1.520690633 Simplifying y + 1.75 = 1.520690633 Reorder the terms: 1.75 + y = 1.520690633 Solving 1.75 + y = 1.520690633 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + y = 1.520690633 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = 1.520690633 + -1.75 y = 1.520690633 + -1.75 Combine like terms: 1.520690633 + -1.75 = -0.229309367 y = -0.229309367 Simplifying y = -0.229309367Subproblem 2
y + 1.75 = -1.520690633 Simplifying y + 1.75 = -1.520690633 Reorder the terms: 1.75 + y = -1.520690633 Solving 1.75 + y = -1.520690633 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + y = -1.520690633 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = -1.520690633 + -1.75 y = -1.520690633 + -1.75 Combine like terms: -1.520690633 + -1.75 = -3.270690633 y = -3.270690633 Simplifying y = -3.270690633Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.229309367, -3.270690633}
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