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Simplifying (3y2 + -5y + 2) + -1(5y2 + -2y + -2) = 0 Reorder the terms: (2 + -5y + 3y2) + -1(5y2 + -2y + -2) = 0 Remove parenthesis around (2 + -5y + 3y2) 2 + -5y + 3y2 + -1(5y2 + -2y + -2) = 0 Reorder the terms: 2 + -5y + 3y2 + -1(-2 + -2y + 5y2) = 0 2 + -5y + 3y2 + (-2 * -1 + -2y * -1 + 5y2 * -1) = 0 2 + -5y + 3y2 + (2 + 2y + -5y2) = 0 Reorder the terms: 2 + 2 + -5y + 2y + 3y2 + -5y2 = 0 Combine like terms: 2 + 2 = 4 4 + -5y + 2y + 3y2 + -5y2 = 0 Combine like terms: -5y + 2y = -3y 4 + -3y + 3y2 + -5y2 = 0 Combine like terms: 3y2 + -5y2 = -2y2 4 + -3y + -2y2 = 0 Solving 4 + -3y + -2y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2 + 1.5y + y2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.5y + 2 + y2 = 0 + 2 Reorder the terms: -2 + 2 + 1.5y + y2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.5y + y2 = 0 + 2 1.5y + y2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.5y + y2 = 2 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 = 2 + 0.5625 Reorder the terms: 0.5625 + 1.5y + y2 = 2 + 0.5625 Combine like terms: 2 + 0.5625 = 2.5625 0.5625 + 1.5y + y2 = 2.5625 Factor a perfect square on the left side: (y + 0.75)(y + 0.75) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (y + 0.75) equal to 1.600781059 and -1.600781059.Subproblem 1
y + 0.75 = 1.600781059 Simplifying y + 0.75 = 1.600781059 Reorder the terms: 0.75 + y = 1.600781059 Solving 0.75 + y = 1.600781059 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 = 1.600781059 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 1.600781059 + -0.75 y = 1.600781059 + -0.75 Combine like terms: 1.600781059 + -0.75 = 0.850781059 y = 0.850781059 Simplifying y = 0.850781059Subproblem 2
y + 0.75 = -1.600781059 Simplifying y + 0.75 = -1.600781059 Reorder the terms: 0.75 + y = -1.600781059 Solving 0.75 + y = -1.600781059 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 = -1.600781059 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -1.600781059 + -0.75 y = -1.600781059 + -0.75 Combine like terms: -1.600781059 + -0.75 = -2.350781059 y = -2.350781059 Simplifying y = -2.350781059Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.850781059, -2.350781059}
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