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Simplifying 15x4 + -15x2 + 1 = 0 Reorder the terms: 1 + -15x2 + 15x4 = 0 Solving 1 + -15x2 + 15x4 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. 0.06666666667 + -1x2 + x4 = 0 Move the constant term to the right: Add '-0.06666666667' to each side of the equation. 0.06666666667 + -1x2 + -0.06666666667 + x4 = 0 + -0.06666666667 Reorder the terms: 0.06666666667 + -0.06666666667 + -1x2 + x4 = 0 + -0.06666666667 Combine like terms: 0.06666666667 + -0.06666666667 = 0.00000000000 0.00000000000 + -1x2 + x4 = 0 + -0.06666666667 -1x2 + x4 = 0 + -0.06666666667 Combine like terms: 0 + -0.06666666667 = -0.06666666667 -1x2 + x4 = -0.06666666667 The x term is -1x2. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x2 + 0.25 + x4 = -0.06666666667 + 0.25 Reorder the terms: 0.25 + -1x2 + x4 = -0.06666666667 + 0.25 Combine like terms: -0.06666666667 + 0.25 = 0.18333333333 0.25 + -1x2 + x4 = 0.18333333333 Factor a perfect square on the left side: (x2 + -0.5)(x2 + -0.5) = 0.18333333333 Calculate the square root of the right side: 0.428174419 Break this problem into two subproblems by setting (x2 + -0.5) equal to 0.428174419 and -0.428174419.Subproblem 1
x2 + -0.5 = 0.428174419 Simplifying x2 + -0.5 = 0.428174419 Reorder the terms: -0.5 + x2 = 0.428174419 Solving -0.5 + x2 = 0.428174419 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x2 = 0.428174419 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = 0.428174419 + 0.5 x2 = 0.428174419 + 0.5 Combine like terms: 0.428174419 + 0.5 = 0.928174419 x2 = 0.928174419 Simplifying x2 = 0.928174419 Take the square root of each side: x = {-0.963418091, 0.963418091}Subproblem 2
x2 + -0.5 = -0.428174419 Simplifying x2 + -0.5 = -0.428174419 Reorder the terms: -0.5 + x2 = -0.428174419 Solving -0.5 + x2 = -0.428174419 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x2 = -0.428174419 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = -0.428174419 + 0.5 x2 = -0.428174419 + 0.5 Combine like terms: -0.428174419 + 0.5 = 0.071825581 x2 = 0.071825581 Simplifying x2 = 0.071825581 Take the square root of each side: x = {-0.26800295, 0.26800295}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.963418091, 0.963418091, -0.26800295, 0.26800295}
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