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Simplifying 0 = 15x4 + -15x2 + 2 Reorder the terms: 0 = 2 + -15x2 + 15x4 Solving 0 = 2 + -15x2 + 15x4 Solving for variable 'x'. Combine like terms: 0 + -2 = -2 -2 + 15x2 + -15x4 = 2 + -15x2 + 15x4 + -2 + 15x2 + -15x4 Reorder the terms: -2 + 15x2 + -15x4 = 2 + -2 + -15x2 + 15x2 + 15x4 + -15x4 Combine like terms: 2 + -2 = 0 -2 + 15x2 + -15x4 = 0 + -15x2 + 15x2 + 15x4 + -15x4 -2 + 15x2 + -15x4 = -15x2 + 15x2 + 15x4 + -15x4 Combine like terms: -15x2 + 15x2 = 0 -2 + 15x2 + -15x4 = 0 + 15x4 + -15x4 -2 + 15x2 + -15x4 = 15x4 + -15x4 Combine like terms: 15x4 + -15x4 = 0 -2 + 15x2 + -15x4 = 0 Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. 0.1333333333 + -1x2 + x4 = 0 Move the constant term to the right: Add '-0.1333333333' to each side of the equation. 0.1333333333 + -1x2 + -0.1333333333 + x4 = 0 + -0.1333333333 Reorder the terms: 0.1333333333 + -0.1333333333 + -1x2 + x4 = 0 + -0.1333333333 Combine like terms: 0.1333333333 + -0.1333333333 = 0.0000000000 0.0000000000 + -1x2 + x4 = 0 + -0.1333333333 -1x2 + x4 = 0 + -0.1333333333 Combine like terms: 0 + -0.1333333333 = -0.1333333333 -1x2 + x4 = -0.1333333333 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 = -0.1333333333 + 0.25 Reorder the terms: 0.25 + -1x2 + x4 = -0.1333333333 + 0.25 Combine like terms: -0.1333333333 + 0.25 = 0.1166666667 0.25 + -1x2 + x4 = 0.1166666667 Factor a perfect square on the left side: (x2 + -0.5)(x2 + -0.5) = 0.1166666667 Calculate the square root of the right side: 0.341565026 Break this problem into two subproblems by setting (x2 + -0.5) equal to 0.341565026 and -0.341565026.Subproblem 1
x2 + -0.5 = 0.341565026 Simplifying x2 + -0.5 = 0.341565026 Reorder the terms: -0.5 + x2 = 0.341565026 Solving -0.5 + x2 = 0.341565026 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 = 0.341565026 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = 0.341565026 + 0.5 x2 = 0.341565026 + 0.5 Combine like terms: 0.341565026 + 0.5 = 0.841565026 x2 = 0.841565026 Simplifying x2 = 0.841565026 Take the square root of each side: x = {-0.917368533, 0.917368533}Subproblem 2
x2 + -0.5 = -0.341565026 Simplifying x2 + -0.5 = -0.341565026 Reorder the terms: -0.5 + x2 = -0.341565026 Solving -0.5 + x2 = -0.341565026 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 = -0.341565026 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = -0.341565026 + 0.5 x2 = -0.341565026 + 0.5 Combine like terms: -0.341565026 + 0.5 = 0.158434974 x2 = 0.158434974 Simplifying x2 = 0.158434974 Take the square root of each side: x = {-0.39803891, 0.39803891}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.917368533, 0.917368533, -0.39803891, 0.39803891}
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