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Simplifying 1 = 2y2 + 7y Reorder the terms: 1 = 7y + 2y2 Solving 1 = 7y + 2y2 Solving for variable 'y'. Reorder the terms: 1 + -7y + -2y2 = 7y + -7y + 2y2 + -2y2 Combine like terms: 7y + -7y = 0 1 + -7y + -2y2 = 0 + 2y2 + -2y2 1 + -7y + -2y2 = 2y2 + -2y2 Combine like terms: 2y2 + -2y2 = 0 1 + -7y + -2y2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + 3.5y + y2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 3.5y + 0.5 + y2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 3.5y + y2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 3.5y + y2 = 0 + 0.5 3.5y + y2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 3.5y + y2 = 0.5 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 = 0.5 + 3.0625 Reorder the terms: 3.0625 + 3.5y + y2 = 0.5 + 3.0625 Combine like terms: 0.5 + 3.0625 = 3.5625 3.0625 + 3.5y + y2 = 3.5625 Factor a perfect square on the left side: (y + 1.75)(y + 1.75) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (y + 1.75) equal to 1.887458609 and -1.887458609.Subproblem 1
y + 1.75 = 1.887458609 Simplifying y + 1.75 = 1.887458609 Reorder the terms: 1.75 + y = 1.887458609 Solving 1.75 + y = 1.887458609 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 = 1.887458609 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = 1.887458609 + -1.75 y = 1.887458609 + -1.75 Combine like terms: 1.887458609 + -1.75 = 0.137458609 y = 0.137458609 Simplifying y = 0.137458609Subproblem 2
y + 1.75 = -1.887458609 Simplifying y + 1.75 = -1.887458609 Reorder the terms: 1.75 + y = -1.887458609 Solving 1.75 + y = -1.887458609 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 = -1.887458609 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + y = -1.887458609 + -1.75 y = -1.887458609 + -1.75 Combine like terms: -1.887458609 + -1.75 = -3.637458609 y = -3.637458609 Simplifying y = -3.637458609Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.137458609, -3.637458609}
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