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Simplifying 2y2 + 4y + 2 + y = 3 Reorder the terms: 2 + 4y + y + 2y2 = 3 Combine like terms: 4y + y = 5y 2 + 5y + 2y2 = 3 Solving 2 + 5y + 2y2 = 3 Solving for variable 'y'. Reorder the terms: 2 + -3 + 5y + 2y2 = 3 + -3 Combine like terms: 2 + -3 = -1 -1 + 5y + 2y2 = 3 + -3 Combine like terms: 3 + -3 = 0 -1 + 5y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + 2.5y + y2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 2.5y + 0.5 + y2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 2.5y + y2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 2.5y + y2 = 0 + 0.5 2.5y + y2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 2.5y + y2 = 0.5 The y term is 2.5y. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5y + 1.5625 + y2 = 0.5 + 1.5625 Reorder the terms: 1.5625 + 2.5y + y2 = 0.5 + 1.5625 Combine like terms: 0.5 + 1.5625 = 2.0625 1.5625 + 2.5y + y2 = 2.0625 Factor a perfect square on the left side: (y + 1.25)(y + 1.25) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (y + 1.25) equal to 1.436140662 and -1.436140662.Subproblem 1
y + 1.25 = 1.436140662 Simplifying y + 1.25 = 1.436140662 Reorder the terms: 1.25 + y = 1.436140662 Solving 1.25 + y = 1.436140662 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + y = 1.436140662 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + y = 1.436140662 + -1.25 y = 1.436140662 + -1.25 Combine like terms: 1.436140662 + -1.25 = 0.186140662 y = 0.186140662 Simplifying y = 0.186140662Subproblem 2
y + 1.25 = -1.436140662 Simplifying y + 1.25 = -1.436140662 Reorder the terms: 1.25 + y = -1.436140662 Solving 1.25 + y = -1.436140662 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + y = -1.436140662 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + y = -1.436140662 + -1.25 y = -1.436140662 + -1.25 Combine like terms: -1.436140662 + -1.25 = -2.686140662 y = -2.686140662 Simplifying y = -2.686140662Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.186140662, -2.686140662}
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