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Simplifying (4y2 + 5y + -1) + -1(2y2 + 3y + 2) = 0 Reorder the terms: (-1 + 5y + 4y2) + -1(2y2 + 3y + 2) = 0 Remove parenthesis around (-1 + 5y + 4y2) -1 + 5y + 4y2 + -1(2y2 + 3y + 2) = 0 Reorder the terms: -1 + 5y + 4y2 + -1(2 + 3y + 2y2) = 0 -1 + 5y + 4y2 + (2 * -1 + 3y * -1 + 2y2 * -1) = 0 -1 + 5y + 4y2 + (-2 + -3y + -2y2) = 0 Reorder the terms: -1 + -2 + 5y + -3y + 4y2 + -2y2 = 0 Combine like terms: -1 + -2 = -3 -3 + 5y + -3y + 4y2 + -2y2 = 0 Combine like terms: 5y + -3y = 2y -3 + 2y + 4y2 + -2y2 = 0 Combine like terms: 4y2 + -2y2 = 2y2 -3 + 2y + 2y2 = 0 Solving -3 + 2y + 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'. -1.5 + y + y2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + y + 1.5 + y2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + y + y2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y + y2 = 0 + 1.5 y + y2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 y + y2 = 1.5 The y term is y. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + y2 = 1.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + y2 = 1.5 + 0.25 Combine like terms: 1.5 + 0.25 = 1.75 1.25 + y2 = 1.75 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (y + 0.5) equal to 1.322875656 and -1.322875656.Subproblem 1
y + 0.5 = 1.322875656 Simplifying y + 0.5 = 1.322875656 Reorder the terms: 0.5 + y = 1.322875656 Solving 0.5 + y = 1.322875656 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 1.322875656 + -0.5 y = 1.322875656 + -0.5 Combine like terms: 1.322875656 + -0.5 = 0.822875656 y = 0.822875656 Simplifying y = 0.822875656Subproblem 2
y + 0.5 = -1.322875656 Simplifying y + 0.5 = -1.322875656 Reorder the terms: 0.5 + y = -1.322875656 Solving 0.5 + y = -1.322875656 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -1.322875656 + -0.5 y = -1.322875656 + -0.5 Combine like terms: -1.322875656 + -0.5 = -1.822875656 y = -1.822875656 Simplifying y = -1.822875656Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.822875656, -1.822875656}
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