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Simplifying 20x4 + -53x2 + 20 = 0 Reorder the terms: 20 + -53x2 + 20x4 = 0 Solving 20 + -53x2 + 20x4 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. 1 + -2.65x2 + x4 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.65x2 + -1 + x4 = 0 + -1 Reorder the terms: 1 + -1 + -2.65x2 + x4 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.65x2 + x4 = 0 + -1 -2.65x2 + x4 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.65x2 + x4 = -1 The x term is -2.65x2. Take half its coefficient (-1.325). Square it (1.755625) and add it to both sides. Add '1.755625' to each side of the equation. -2.65x2 + 1.755625 + x4 = -1 + 1.755625 Reorder the terms: 1.755625 + -2.65x2 + x4 = -1 + 1.755625 Combine like terms: -1 + 1.755625 = 0.755625 1.755625 + -2.65x2 + x4 = 0.755625 Factor a perfect square on the left side: (x2 + -1.325)(x2 + -1.325) = 0.755625 Calculate the square root of the right side: 0.869266933 Break this problem into two subproblems by setting (x2 + -1.325) equal to 0.869266933 and -0.869266933.Subproblem 1
x2 + -1.325 = 0.869266933 Simplifying x2 + -1.325 = 0.869266933 Reorder the terms: -1.325 + x2 = 0.869266933 Solving -1.325 + x2 = 0.869266933 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.325' to each side of the equation. -1.325 + 1.325 + x2 = 0.869266933 + 1.325 Combine like terms: -1.325 + 1.325 = 0.000 0.000 + x2 = 0.869266933 + 1.325 x2 = 0.869266933 + 1.325 Combine like terms: 0.869266933 + 1.325 = 2.194266933 x2 = 2.194266933 Simplifying x2 = 2.194266933 Take the square root of each side: x = {-1.48130582, 1.48130582}Subproblem 2
x2 + -1.325 = -0.869266933 Simplifying x2 + -1.325 = -0.869266933 Reorder the terms: -1.325 + x2 = -0.869266933 Solving -1.325 + x2 = -0.869266933 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.325' to each side of the equation. -1.325 + 1.325 + x2 = -0.869266933 + 1.325 Combine like terms: -1.325 + 1.325 = 0.000 0.000 + x2 = -0.869266933 + 1.325 x2 = -0.869266933 + 1.325 Combine like terms: -0.869266933 + 1.325 = 0.455733067 x2 = 0.455733067 Simplifying x2 = 0.455733067 Take the square root of each side: x = {-0.675080045, 0.675080045}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.48130582, 1.48130582, -0.675080045, 0.675080045}
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