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Simplifying 0 = x4 + -8x2 + 8 Reorder the terms: 0 = 8 + -8x2 + x4 Solving 0 = 8 + -8x2 + x4 Solving for variable 'x'. Combine like terms: 0 + -8 = -8 -8 + 8x2 + -1x4 = 8 + -8x2 + x4 + -8 + 8x2 + -1x4 Reorder the terms: -8 + 8x2 + -1x4 = 8 + -8 + -8x2 + 8x2 + x4 + -1x4 Combine like terms: 8 + -8 = 0 -8 + 8x2 + -1x4 = 0 + -8x2 + 8x2 + x4 + -1x4 -8 + 8x2 + -1x4 = -8x2 + 8x2 + x4 + -1x4 Combine like terms: -8x2 + 8x2 = 0 -8 + 8x2 + -1x4 = 0 + x4 + -1x4 -8 + 8x2 + -1x4 = x4 + -1x4 Combine like terms: x4 + -1x4 = 0 -8 + 8x2 + -1x4 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 8 + -8x2 + x4 = 0 Move the constant term to the right: Add '-8' to each side of the equation. 8 + -8x2 + -8 + x4 = 0 + -8 Reorder the terms: 8 + -8 + -8x2 + x4 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + -8x2 + x4 = 0 + -8 -8x2 + x4 = 0 + -8 Combine like terms: 0 + -8 = -8 -8x2 + x4 = -8 The x term is -8x2. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8x2 + 16 + x4 = -8 + 16 Reorder the terms: 16 + -8x2 + x4 = -8 + 16 Combine like terms: -8 + 16 = 8 16 + -8x2 + x4 = 8 Factor a perfect square on the left side: (x2 + -4)(x2 + -4) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (x2 + -4) equal to 2.828427125 and -2.828427125.Subproblem 1
x2 + -4 = 2.828427125 Simplifying x2 + -4 = 2.828427125 Reorder the terms: -4 + x2 = 2.828427125 Solving -4 + x2 = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = 2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = 2.828427125 + 4 x2 = 2.828427125 + 4 Combine like terms: 2.828427125 + 4 = 6.828427125 x2 = 6.828427125 Simplifying x2 = 6.828427125 Take the square root of each side: x = {-2.61312593, 2.61312593}Subproblem 2
x2 + -4 = -2.828427125 Simplifying x2 + -4 = -2.828427125 Reorder the terms: -4 + x2 = -2.828427125 Solving -4 + x2 = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = -2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = -2.828427125 + 4 x2 = -2.828427125 + 4 Combine like terms: -2.828427125 + 4 = 1.171572875 x2 = 1.171572875 Simplifying x2 = 1.171572875 Take the square root of each side: x = {-1.0823922, 1.0823922}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.61312593, 2.61312593, -1.0823922, 1.0823922}
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