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Simplifying 4x(x) + 1.41421356237x + -1 = 0 Multiply x * x 4x2 + 1.41421356237x + -1 = 0 Reorder the terms: -1 + 1.41421356237x + 4x2 = 0 Solving -1 + 1.41421356237x + 4x2 = 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'. -0.25 + 0.3535533906x + x2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 0.3535533906x + 0.25 + x2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 0.3535533906x + x2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 0.3535533906x + x2 = 0 + 0.25 0.3535533906x + x2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 0.3535533906x + x2 = 0.25 The x term is 0.3535533906x. Take half its coefficient (0.1767766953). Square it (0.03125000000) and add it to both sides. Add '0.03125000000' to each side of the equation. 0.3535533906x + 0.03125000000 + x2 = 0.25 + 0.03125000000 Reorder the terms: 0.03125000000 + 0.3535533906x + x2 = 0.25 + 0.03125000000 Combine like terms: 0.25 + 0.03125000000 = 0.28125 0.03125000000 + 0.3535533906x + x2 = 0.28125 Factor a perfect square on the left side: (x + 0.1767766953)(x + 0.1767766953) = 0.28125 Calculate the square root of the right side: 0.530330086 Break this problem into two subproblems by setting (x + 0.1767766953) equal to 0.530330086 and -0.530330086.Subproblem 1
x + 0.1767766953 = 0.530330086 Simplifying x + 0.1767766953 = 0.530330086 Reorder the terms: 0.1767766953 + x = 0.530330086 Solving 0.1767766953 + x = 0.530330086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1767766953' to each side of the equation. 0.1767766953 + -0.1767766953 + x = 0.530330086 + -0.1767766953 Combine like terms: 0.1767766953 + -0.1767766953 = 0.0000000000 0.0000000000 + x = 0.530330086 + -0.1767766953 x = 0.530330086 + -0.1767766953 Combine like terms: 0.530330086 + -0.1767766953 = 0.3535533907 x = 0.3535533907 Simplifying x = 0.3535533907Subproblem 2
x + 0.1767766953 = -0.530330086 Simplifying x + 0.1767766953 = -0.530330086 Reorder the terms: 0.1767766953 + x = -0.530330086 Solving 0.1767766953 + x = -0.530330086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1767766953' to each side of the equation. 0.1767766953 + -0.1767766953 + x = -0.530330086 + -0.1767766953 Combine like terms: 0.1767766953 + -0.1767766953 = 0.0000000000 0.0000000000 + x = -0.530330086 + -0.1767766953 x = -0.530330086 + -0.1767766953 Combine like terms: -0.530330086 + -0.1767766953 = -0.7071067813 x = -0.7071067813 Simplifying x = -0.7071067813Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.3535533907, -0.7071067813}
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