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Simplifying 4x4 + -16.25x2 + 4 = 0 Reorder the terms: 4 + -16.25x2 + 4x4 = 0 Solving 4 + -16.25x2 + 4x4 = 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'. 1 + -4.0625x2 + x4 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -4.0625x2 + -1 + x4 = 0 + -1 Reorder the terms: 1 + -1 + -4.0625x2 + x4 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -4.0625x2 + x4 = 0 + -1 -4.0625x2 + x4 = 0 + -1 Combine like terms: 0 + -1 = -1 -4.0625x2 + x4 = -1 The x term is -4.0625x2. Take half its coefficient (-2.03125). Square it (4.125976563) and add it to both sides. Add '4.125976563' to each side of the equation. -4.0625x2 + 4.125976563 + x4 = -1 + 4.125976563 Reorder the terms: 4.125976563 + -4.0625x2 + x4 = -1 + 4.125976563 Combine like terms: -1 + 4.125976563 = 3.125976563 4.125976563 + -4.0625x2 + x4 = 3.125976563 Factor a perfect square on the left side: (x2 + -2.03125)(x2 + -2.03125) = 3.125976563 Calculate the square root of the right side: 1.768043145 Break this problem into two subproblems by setting (x2 + -2.03125) equal to 1.768043145 and -1.768043145.Subproblem 1
x2 + -2.03125 = 1.768043145 Simplifying x2 + -2.03125 = 1.768043145 Reorder the terms: -2.03125 + x2 = 1.768043145 Solving -2.03125 + x2 = 1.768043145 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.03125' to each side of the equation. -2.03125 + 2.03125 + x2 = 1.768043145 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + x2 = 1.768043145 + 2.03125 x2 = 1.768043145 + 2.03125 Combine like terms: 1.768043145 + 2.03125 = 3.799293145 x2 = 3.799293145 Simplifying x2 = 3.799293145 Take the square root of each side: x = {-1.949177556, 1.949177556}Subproblem 2
x2 + -2.03125 = -1.768043145 Simplifying x2 + -2.03125 = -1.768043145 Reorder the terms: -2.03125 + x2 = -1.768043145 Solving -2.03125 + x2 = -1.768043145 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.03125' to each side of the equation. -2.03125 + 2.03125 + x2 = -1.768043145 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + x2 = -1.768043145 + 2.03125 x2 = -1.768043145 + 2.03125 Combine like terms: -1.768043145 + 2.03125 = 0.263206855 x2 = 0.263206855 Simplifying x2 = 0.263206855 Take the square root of each side: x = {-0.513036894, 0.513036894}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.949177556, 1.949177556, -0.513036894, 0.513036894}
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