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Simplifying 4y2 + -12y = -1 Reorder the terms: -12y + 4y2 = -1 Solving -12y + 4y2 = -1 Solving for variable 'y'. Reorder the terms: 1 + -12y + 4y2 = -1 + 1 Combine like terms: -1 + 1 = 0 1 + -12y + 4y2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + -3y + y2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -3y + -0.25 + y2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -3y + y2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -3y + y2 = 0 + -0.25 -3y + y2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -3y + y2 = -0.25 The y term is -3y. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3y + 2.25 + y2 = -0.25 + 2.25 Reorder the terms: 2.25 + -3y + y2 = -0.25 + 2.25 Combine like terms: -0.25 + 2.25 = 2 2.25 + -3y + y2 = 2 Factor a perfect square on the left side: (y + -1.5)(y + -1.5) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (y + -1.5) equal to 1.414213562 and -1.414213562.Subproblem 1
y + -1.5 = 1.414213562 Simplifying y + -1.5 = 1.414213562 Reorder the terms: -1.5 + y = 1.414213562 Solving -1.5 + y = 1.414213562 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + y = 1.414213562 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = 1.414213562 + 1.5 y = 1.414213562 + 1.5 Combine like terms: 1.414213562 + 1.5 = 2.914213562 y = 2.914213562 Simplifying y = 2.914213562Subproblem 2
y + -1.5 = -1.414213562 Simplifying y + -1.5 = -1.414213562 Reorder the terms: -1.5 + y = -1.414213562 Solving -1.5 + y = -1.414213562 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + y = -1.414213562 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = -1.414213562 + 1.5 y = -1.414213562 + 1.5 Combine like terms: -1.414213562 + 1.5 = 0.085786438 y = 0.085786438 Simplifying y = 0.085786438Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.914213562, 0.085786438}
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