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Simplifying 2y2 + 11y + 3 = 0 Reorder the terms: 3 + 11y + 2y2 = 0 Solving 3 + 11y + 2y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + 5.5y + y2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 5.5y + -1.5 + y2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 5.5y + y2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 5.5y + y2 = 0 + -1.5 5.5y + y2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 5.5y + y2 = -1.5 The y term is 5.5y. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5y + 7.5625 + y2 = -1.5 + 7.5625 Reorder the terms: 7.5625 + 5.5y + y2 = -1.5 + 7.5625 Combine like terms: -1.5 + 7.5625 = 6.0625 7.5625 + 5.5y + y2 = 6.0625 Factor a perfect square on the left side: (y + 2.75)(y + 2.75) = 6.0625 Calculate the square root of the right side: 2.46221445 Break this problem into two subproblems by setting (y + 2.75) equal to 2.46221445 and -2.46221445.Subproblem 1
y + 2.75 = 2.46221445 Simplifying y + 2.75 = 2.46221445 Reorder the terms: 2.75 + y = 2.46221445 Solving 2.75 + y = 2.46221445 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + y = 2.46221445 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + y = 2.46221445 + -2.75 y = 2.46221445 + -2.75 Combine like terms: 2.46221445 + -2.75 = -0.28778555 y = -0.28778555 Simplifying y = -0.28778555Subproblem 2
y + 2.75 = -2.46221445 Simplifying y + 2.75 = -2.46221445 Reorder the terms: 2.75 + y = -2.46221445 Solving 2.75 + y = -2.46221445 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + y = -2.46221445 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + y = -2.46221445 + -2.75 y = -2.46221445 + -2.75 Combine like terms: -2.46221445 + -2.75 = -5.21221445 y = -5.21221445 Simplifying y = -5.21221445Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.28778555, -5.21221445}
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