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Simplifying 2y2 + 14y = -19 Reorder the terms: 14y + 2y2 = -19 Solving 14y + 2y2 = -19 Solving for variable 'y'. Reorder the terms: 19 + 14y + 2y2 = -19 + 19 Combine like terms: -19 + 19 = 0 19 + 14y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 9.5 + 7y + y2 = 0 Move the constant term to the right: Add '-9.5' to each side of the equation. 9.5 + 7y + -9.5 + y2 = 0 + -9.5 Reorder the terms: 9.5 + -9.5 + 7y + y2 = 0 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + 7y + y2 = 0 + -9.5 7y + y2 = 0 + -9.5 Combine like terms: 0 + -9.5 = -9.5 7y + y2 = -9.5 The y term is 7y. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7y + 12.25 + y2 = -9.5 + 12.25 Reorder the terms: 12.25 + 7y + y2 = -9.5 + 12.25 Combine like terms: -9.5 + 12.25 = 2.75 12.25 + 7y + y2 = 2.75 Factor a perfect square on the left side: (y + 3.5)(y + 3.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (y + 3.5) equal to 1.658312395 and -1.658312395.Subproblem 1
y + 3.5 = 1.658312395 Simplifying y + 3.5 = 1.658312395 Reorder the terms: 3.5 + y = 1.658312395 Solving 3.5 + y = 1.658312395 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + y = 1.658312395 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + y = 1.658312395 + -3.5 y = 1.658312395 + -3.5 Combine like terms: 1.658312395 + -3.5 = -1.841687605 y = -1.841687605 Simplifying y = -1.841687605Subproblem 2
y + 3.5 = -1.658312395 Simplifying y + 3.5 = -1.658312395 Reorder the terms: 3.5 + y = -1.658312395 Solving 3.5 + y = -1.658312395 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + y = -1.658312395 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + y = -1.658312395 + -3.5 y = -1.658312395 + -3.5 Combine like terms: -1.658312395 + -3.5 = -5.158312395 y = -5.158312395 Simplifying y = -5.158312395Solution
The solution to the problem is based on the solutions from the subproblems. y = {-1.841687605, -5.158312395}
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