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Simplifying 2y2 + 3y + 5 = 8 Reorder the terms: 5 + 3y + 2y2 = 8 Solving 5 + 3y + 2y2 = 8 Solving for variable 'y'. Reorder the terms: 5 + -8 + 3y + 2y2 = 8 + -8 Combine like terms: 5 + -8 = -3 -3 + 3y + 2y2 = 8 + -8 Combine like terms: 8 + -8 = 0 -3 + 3y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + 1.5y + y2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 1.5y + 1.5 + y2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 1.5y + y2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 1.5y + y2 = 0 + 1.5 1.5y + y2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 1.5y + y2 = 1.5 The y term is 1.5y. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5y + 0.5625 + y2 = 1.5 + 0.5625 Reorder the terms: 0.5625 + 1.5y + y2 = 1.5 + 0.5625 Combine like terms: 1.5 + 0.5625 = 2.0625 0.5625 + 1.5y + y2 = 2.0625 Factor a perfect square on the left side: (y + 0.75)(y + 0.75) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (y + 0.75) equal to 1.436140662 and -1.436140662.Subproblem 1
y + 0.75 = 1.436140662 Simplifying y + 0.75 = 1.436140662 Reorder the terms: 0.75 + y = 1.436140662 Solving 0.75 + y = 1.436140662 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = 1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 1.436140662 + -0.75 y = 1.436140662 + -0.75 Combine like terms: 1.436140662 + -0.75 = 0.686140662 y = 0.686140662 Simplifying y = 0.686140662Subproblem 2
y + 0.75 = -1.436140662 Simplifying y + 0.75 = -1.436140662 Reorder the terms: 0.75 + y = -1.436140662 Solving 0.75 + y = -1.436140662 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = -1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -1.436140662 + -0.75 y = -1.436140662 + -0.75 Combine like terms: -1.436140662 + -0.75 = -2.186140662 y = -2.186140662 Simplifying y = -2.186140662Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.686140662, -2.186140662}
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