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Simplifying 2x2 + x * y + -2y2 = 0 Multiply x * y 2x2 + xy + -2y2 = 0 Reorder the terms: xy + 2x2 + -2y2 = 0 Solving xy + 2x2 + -2y2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5xy + x2 + -1y2 = 0 Move the constant term to the right: Add 'y2' to each side of the equation. 0.5xy + x2 + -1y2 + y2 = 0 + y2 Combine like terms: -1y2 + y2 = 0 0.5xy + x2 + 0 = 0 + y2 0.5xy + x2 = 0 + y2 Remove the zero: 0.5xy + x2 = y2 The x term is xy. Take half its coefficient (0.5y). Square it (0.25y2) and add it to both sides. Add '0.25y2' to each side of the equation. 0.5xy + x2 + 0.25y2 = y2 + 0.25y2 Combine like terms: y2 + 0.25y2 = 1.25y2 0.5xy + x2 + 0.25y2 = 1.25y2 Factor a perfect square on the left side: (x + 0.5y)(x + 0.5y) = 1.25y2 Calculate the square root of the right side: 1.118033989y Break this problem into two subproblems by setting (x + 0.5y) equal to 1.118033989y and -1.118033989y.Subproblem 1
x + 0.5y = 1.118033989y Simplifying x + 0.5y = 1.118033989y Solving x + 0.5y = 1.118033989y Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5y' to each side of the equation. x + 0.5y + -0.5y = 1.118033989y + -0.5y Combine like terms: 0.5y + -0.5y = 0.0 x + 0.0 = 1.118033989y + -0.5y x = 1.118033989y + -0.5y Combine like terms: 1.118033989y + -0.5y = 0.618033989y x = 0.618033989y Simplifying x = 0.618033989ySubproblem 2
x + 0.5y = -1.118033989y Simplifying x + 0.5y = -1.118033989y Solving x + 0.5y = -1.118033989y Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5y' to each side of the equation. x + 0.5y + -0.5y = -1.118033989y + -0.5y Combine like terms: 0.5y + -0.5y = 0.0 x + 0.0 = -1.118033989y + -0.5y x = -1.118033989y + -0.5y Combine like terms: -1.118033989y + -0.5y = -1.618033989y x = -1.618033989y Simplifying x = -1.618033989ySolution
The solution to the problem is based on the solutions from the subproblems. x = {0.618033989y, -1.618033989y}
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