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Simplifying 4y2 + 4y = 1 Reorder the terms: 4y + 4y2 = 1 Solving 4y + 4y2 = 1 Solving for variable 'y'. Reorder the terms: -1 + 4y + 4y2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 4y + 4y2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + y + y2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + y + 0.25 + y2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + y + y2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y + y2 = 0 + 0.25 y + y2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 y + y2 = 0.25 The y term is y. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + y2 = 0.25 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + y2 = 0.25 + 0.25 Combine like terms: 0.25 + 0.25 = 0.5 1.25 + y2 = 0.5 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (y + 0.5) equal to 0.707106781 and -0.707106781.Subproblem 1
y + 0.5 = 0.707106781 Simplifying y + 0.5 = 0.707106781 Reorder the terms: 0.5 + y = 0.707106781 Solving 0.5 + y = 0.707106781 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 0.707106781 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 0.707106781 + -0.5 y = 0.707106781 + -0.5 Combine like terms: 0.707106781 + -0.5 = 0.207106781 y = 0.207106781 Simplifying y = 0.207106781Subproblem 2
y + 0.5 = -0.707106781 Simplifying y + 0.5 = -0.707106781 Reorder the terms: 0.5 + y = -0.707106781 Solving 0.5 + y = -0.707106781 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -0.707106781 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -0.707106781 + -0.5 y = -0.707106781 + -0.5 Combine like terms: -0.707106781 + -0.5 = -1.207106781 y = -1.207106781 Simplifying y = -1.207106781Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.207106781, -1.207106781}
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