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Simplifying 6y2 + 3y = 5 Reorder the terms: 3y + 6y2 = 5 Solving 3y + 6y2 = 5 Solving for variable 'y'. Reorder the terms: -5 + 3y + 6y2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + 3y + 6y2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.8333333333 + 0.5y + y2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + 0.5y + 0.8333333333 + y2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + 0.5y + y2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + 0.5y + y2 = 0 + 0.8333333333 0.5y + y2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 0.5y + y2 = 0.8333333333 The y term is 0.5y. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5y + 0.0625 + y2 = 0.8333333333 + 0.0625 Reorder the terms: 0.0625 + 0.5y + y2 = 0.8333333333 + 0.0625 Combine like terms: 0.8333333333 + 0.0625 = 0.8958333333 0.0625 + 0.5y + y2 = 0.8958333333 Factor a perfect square on the left side: (y + 0.25)(y + 0.25) = 0.8958333333 Calculate the square root of the right side: 0.946484724 Break this problem into two subproblems by setting (y + 0.25) equal to 0.946484724 and -0.946484724.Subproblem 1
y + 0.25 = 0.946484724 Simplifying y + 0.25 = 0.946484724 Reorder the terms: 0.25 + y = 0.946484724 Solving 0.25 + y = 0.946484724 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + y = 0.946484724 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + y = 0.946484724 + -0.25 y = 0.946484724 + -0.25 Combine like terms: 0.946484724 + -0.25 = 0.696484724 y = 0.696484724 Simplifying y = 0.696484724Subproblem 2
y + 0.25 = -0.946484724 Simplifying y + 0.25 = -0.946484724 Reorder the terms: 0.25 + y = -0.946484724 Solving 0.25 + y = -0.946484724 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + y = -0.946484724 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + y = -0.946484724 + -0.25 y = -0.946484724 + -0.25 Combine like terms: -0.946484724 + -0.25 = -1.196484724 y = -1.196484724 Simplifying y = -1.196484724Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.696484724, -1.196484724}
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