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Simplifying 16x4 + -81x2 + 1 = 0 Reorder the terms: 1 + -81x2 + 16x4 = 0 Solving 1 + -81x2 + 16x4 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.0625 + -5.0625x2 + x4 = 0 Move the constant term to the right: Add '-0.0625' to each side of the equation. 0.0625 + -5.0625x2 + -0.0625 + x4 = 0 + -0.0625 Reorder the terms: 0.0625 + -0.0625 + -5.0625x2 + x4 = 0 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + -5.0625x2 + x4 = 0 + -0.0625 -5.0625x2 + x4 = 0 + -0.0625 Combine like terms: 0 + -0.0625 = -0.0625 -5.0625x2 + x4 = -0.0625 The x term is -5.0625x2. Take half its coefficient (-2.53125). Square it (6.407226563) and add it to both sides. Add '6.407226563' to each side of the equation. -5.0625x2 + 6.407226563 + x4 = -0.0625 + 6.407226563 Reorder the terms: 6.407226563 + -5.0625x2 + x4 = -0.0625 + 6.407226563 Combine like terms: -0.0625 + 6.407226563 = 6.344726563 6.407226563 + -5.0625x2 + x4 = 6.344726563 Factor a perfect square on the left side: (x2 + -2.53125)(x2 + -2.53125) = 6.344726563 Calculate the square root of the right side: 2.518874067 Break this problem into two subproblems by setting (x2 + -2.53125) equal to 2.518874067 and -2.518874067.Subproblem 1
x2 + -2.53125 = 2.518874067 Simplifying x2 + -2.53125 = 2.518874067 Reorder the terms: -2.53125 + x2 = 2.518874067 Solving -2.53125 + x2 = 2.518874067 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.53125' to each side of the equation. -2.53125 + 2.53125 + x2 = 2.518874067 + 2.53125 Combine like terms: -2.53125 + 2.53125 = 0.00000 0.00000 + x2 = 2.518874067 + 2.53125 x2 = 2.518874067 + 2.53125 Combine like terms: 2.518874067 + 2.53125 = 5.050124067 x2 = 5.050124067 Simplifying x2 = 5.050124067 Take the square root of each side: x = {-2.24724811, 2.24724811}Subproblem 2
x2 + -2.53125 = -2.518874067 Simplifying x2 + -2.53125 = -2.518874067 Reorder the terms: -2.53125 + x2 = -2.518874067 Solving -2.53125 + x2 = -2.518874067 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.53125' to each side of the equation. -2.53125 + 2.53125 + x2 = -2.518874067 + 2.53125 Combine like terms: -2.53125 + 2.53125 = 0.00000 0.00000 + x2 = -2.518874067 + 2.53125 x2 = -2.518874067 + 2.53125 Combine like terms: -2.518874067 + 2.53125 = 0.012375933 x2 = 0.012375933 Simplifying x2 = 0.012375933 Take the square root of each side: x = {-0.111247171, 0.111247171}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.24724811, 2.24724811, -0.111247171, 0.111247171}
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