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Simplifying 3(-2 + y) + 2y2 = 12 (-2 * 3 + y * 3) + 2y2 = 12 (-6 + 3y) + 2y2 = 12 Solving -6 + 3y + 2y2 = 12 Solving for variable 'y'. Reorder the terms: -6 + -12 + 3y + 2y2 = 12 + -12 Combine like terms: -6 + -12 = -18 -18 + 3y + 2y2 = 12 + -12 Combine like terms: 12 + -12 = 0 -18 + 3y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9 + 1.5y + y2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 1.5y + 9 + y2 = 0 + 9 Reorder the terms: -9 + 9 + 1.5y + y2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 1.5y + y2 = 0 + 9 1.5y + y2 = 0 + 9 Combine like terms: 0 + 9 = 9 1.5y + y2 = 9 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 = 9 + 0.5625 Reorder the terms: 0.5625 + 1.5y + y2 = 9 + 0.5625 Combine like terms: 9 + 0.5625 = 9.5625 0.5625 + 1.5y + y2 = 9.5625 Factor a perfect square on the left side: (y + 0.75)(y + 0.75) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (y + 0.75) equal to 3.092329219 and -3.092329219.Subproblem 1
y + 0.75 = 3.092329219 Simplifying y + 0.75 = 3.092329219 Reorder the terms: 0.75 + y = 3.092329219 Solving 0.75 + y = 3.092329219 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 = 3.092329219 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 3.092329219 + -0.75 y = 3.092329219 + -0.75 Combine like terms: 3.092329219 + -0.75 = 2.342329219 y = 2.342329219 Simplifying y = 2.342329219Subproblem 2
y + 0.75 = -3.092329219 Simplifying y + 0.75 = -3.092329219 Reorder the terms: 0.75 + y = -3.092329219 Solving 0.75 + y = -3.092329219 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 = -3.092329219 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -3.092329219 + -0.75 y = -3.092329219 + -0.75 Combine like terms: -3.092329219 + -0.75 = -3.842329219 y = -3.842329219 Simplifying y = -3.842329219Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.342329219, -3.842329219}
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