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Simplifying 4x2 + 4xy + -1y2 = 0 Reorder the terms: 4xy + 4x2 + -1y2 = 0 Solving 4xy + 4x2 + -1y2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. xy + x2 + -0.25y2 = 0 Move the constant term to the right: Add '0.25y2' to each side of the equation. xy + x2 + -0.25y2 + 0.25y2 = 0 + 0.25y2 Combine like terms: -0.25y2 + 0.25y2 = 0.00 xy + x2 + 0.00 = 0 + 0.25y2 xy + x2 = 0 + 0.25y2 Remove the zero: xy + x2 = 0.25y2 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. y2 + x2 + 0.25y2 = 0.25y2 + 0.25y2 Reorder the terms: x2 + y2 + 0.25y2 = 0.25y2 + 0.25y2 Combine like terms: y2 + 0.25y2 = 1.25y2 x2 + 1.25y2 = 0.25y2 + 0.25y2 Combine like terms: 0.25y2 + 0.25y2 = 0.5y2 x2 + 1.25y2 = 0.5y2 Factor a perfect square on the left side: (x + 0.5y)(x + 0.5y) = 0.5y2 Calculate the square root of the right side: 0.707106781y Break this problem into two subproblems by setting (x + 0.5y) equal to 0.707106781y and -0.707106781y.Subproblem 1
x + 0.5y = 0.707106781y Simplifying x + 0.5y = 0.707106781y Solving x + 0.5y = 0.707106781y 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 = 0.707106781y + -0.5y Combine like terms: 0.5y + -0.5y = 0.0 x + 0.0 = 0.707106781y + -0.5y x = 0.707106781y + -0.5y Combine like terms: 0.707106781y + -0.5y = 0.207106781y x = 0.207106781y Simplifying x = 0.207106781ySubproblem 2
x + 0.5y = -0.707106781y Simplifying x + 0.5y = -0.707106781y Solving x + 0.5y = -0.707106781y 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 = -0.707106781y + -0.5y Combine like terms: 0.5y + -0.5y = 0.0 x + 0.0 = -0.707106781y + -0.5y x = -0.707106781y + -0.5y Combine like terms: -0.707106781y + -0.5y = -1.207106781y x = -1.207106781y Simplifying x = -1.207106781ySolution
The solution to the problem is based on the solutions from the subproblems. x = {0.207106781y, -1.207106781y}
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