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Simplifying 0 = x4 + -109x2 + 54 Reorder the terms: 0 = 54 + -109x2 + x4 Solving 0 = 54 + -109x2 + x4 Solving for variable 'x'. Combine like terms: 0 + -54 = -54 -54 + 109x2 + -1x4 = 54 + -109x2 + x4 + -54 + 109x2 + -1x4 Reorder the terms: -54 + 109x2 + -1x4 = 54 + -54 + -109x2 + 109x2 + x4 + -1x4 Combine like terms: 54 + -54 = 0 -54 + 109x2 + -1x4 = 0 + -109x2 + 109x2 + x4 + -1x4 -54 + 109x2 + -1x4 = -109x2 + 109x2 + x4 + -1x4 Combine like terms: -109x2 + 109x2 = 0 -54 + 109x2 + -1x4 = 0 + x4 + -1x4 -54 + 109x2 + -1x4 = x4 + -1x4 Combine like terms: x4 + -1x4 = 0 -54 + 109x2 + -1x4 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 54 + -109x2 + x4 = 0 Move the constant term to the right: Add '-54' to each side of the equation. 54 + -109x2 + -54 + x4 = 0 + -54 Reorder the terms: 54 + -54 + -109x2 + x4 = 0 + -54 Combine like terms: 54 + -54 = 0 0 + -109x2 + x4 = 0 + -54 -109x2 + x4 = 0 + -54 Combine like terms: 0 + -54 = -54 -109x2 + x4 = -54 The x term is -109x2. Take half its coefficient (-54.5). Square it (2970.25) and add it to both sides. Add '2970.25' to each side of the equation. -109x2 + 2970.25 + x4 = -54 + 2970.25 Reorder the terms: 2970.25 + -109x2 + x4 = -54 + 2970.25 Combine like terms: -54 + 2970.25 = 2916.25 2970.25 + -109x2 + x4 = 2916.25 Factor a perfect square on the left side: (x2 + -54.5)(x2 + -54.5) = 2916.25 Calculate the square root of the right side: 54.002314765 Break this problem into two subproblems by setting (x2 + -54.5) equal to 54.002314765 and -54.002314765.Subproblem 1
x2 + -54.5 = 54.002314765 Simplifying x2 + -54.5 = 54.002314765 Reorder the terms: -54.5 + x2 = 54.002314765 Solving -54.5 + x2 = 54.002314765 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '54.5' to each side of the equation. -54.5 + 54.5 + x2 = 54.002314765 + 54.5 Combine like terms: -54.5 + 54.5 = 0.0 0.0 + x2 = 54.002314765 + 54.5 x2 = 54.002314765 + 54.5 Combine like terms: 54.002314765 + 54.5 = 108.502314765 x2 = 108.502314765 Simplifying x2 = 108.502314765 Take the square root of each side: x = {-10.41644444, 10.41644444}Subproblem 2
x2 + -54.5 = -54.002314765 Simplifying x2 + -54.5 = -54.002314765 Reorder the terms: -54.5 + x2 = -54.002314765 Solving -54.5 + x2 = -54.002314765 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '54.5' to each side of the equation. -54.5 + 54.5 + x2 = -54.002314765 + 54.5 Combine like terms: -54.5 + 54.5 = 0.0 0.0 + x2 = -54.002314765 + 54.5 x2 = -54.002314765 + 54.5 Combine like terms: -54.002314765 + 54.5 = 0.497685235 x2 = 0.497685235 Simplifying x2 = 0.497685235 Take the square root of each side: x = {-0.705468096, 0.705468096}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-10.41644444, 10.41644444, -0.705468096, 0.705468096}
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