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Simplifying 1744x4 + -75168x2 + 314928 = 0 Reorder the terms: 314928 + -75168x2 + 1744x4 = 0 Solving 314928 + -75168x2 + 1744x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '16'. 16(19683 + -4698x2 + 109x4) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(19683 + -4698x2 + 109x4)' equal to zero and attempt to solve: Simplifying 19683 + -4698x2 + 109x4 = 0 Solving 19683 + -4698x2 + 109x4 = 0 Begin completing the square. Divide all terms by 109 the coefficient of the squared term: Divide each side by '109'. 180.5779817 + -43.10091743x2 + x4 = 0 Move the constant term to the right: Add '-180.5779817' to each side of the equation. 180.5779817 + -43.10091743x2 + -180.5779817 + x4 = 0 + -180.5779817 Reorder the terms: 180.5779817 + -180.5779817 + -43.10091743x2 + x4 = 0 + -180.5779817 Combine like terms: 180.5779817 + -180.5779817 = 0.0000000 0.0000000 + -43.10091743x2 + x4 = 0 + -180.5779817 -43.10091743x2 + x4 = 0 + -180.5779817 Combine like terms: 0 + -180.5779817 = -180.5779817 -43.10091743x2 + x4 = -180.5779817 The x term is -43.10091743x2. Take half its coefficient (-21.55045872). Square it (464.4222710) and add it to both sides. Add '464.4222710' to each side of the equation. -43.10091743x2 + 464.4222710 + x4 = -180.5779817 + 464.4222710 Reorder the terms: 464.4222710 + -43.10091743x2 + x4 = -180.5779817 + 464.4222710 Combine like terms: -180.5779817 + 464.4222710 = 283.8442893 464.4222710 + -43.10091743x2 + x4 = 283.8442893 Factor a perfect square on the left side: (x2 + -21.55045872)(x2 + -21.55045872) = 283.8442893 Calculate the square root of the right side: 16.847679048 Break this problem into two subproblems by setting (x2 + -21.55045872) equal to 16.847679048 and -16.847679048.Subproblem 1
x2 + -21.55045872 = 16.847679048 Simplifying x2 + -21.55045872 = 16.847679048 Reorder the terms: -21.55045872 + x2 = 16.847679048 Solving -21.55045872 + x2 = 16.847679048 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '21.55045872' to each side of the equation. -21.55045872 + 21.55045872 + x2 = 16.847679048 + 21.55045872 Combine like terms: -21.55045872 + 21.55045872 = 0.00000000 0.00000000 + x2 = 16.847679048 + 21.55045872 x2 = 16.847679048 + 21.55045872 Combine like terms: 16.847679048 + 21.55045872 = 38.398137768 x2 = 38.398137768 Simplifying x2 = 38.398137768 Take the square root of each side: x = {-6.196623094, 6.196623094}Subproblem 2
x2 + -21.55045872 = -16.847679048 Simplifying x2 + -21.55045872 = -16.847679048 Reorder the terms: -21.55045872 + x2 = -16.847679048 Solving -21.55045872 + x2 = -16.847679048 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '21.55045872' to each side of the equation. -21.55045872 + 21.55045872 + x2 = -16.847679048 + 21.55045872 Combine like terms: -21.55045872 + 21.55045872 = 0.00000000 0.00000000 + x2 = -16.847679048 + 21.55045872 x2 = -16.847679048 + 21.55045872 Combine like terms: -16.847679048 + 21.55045872 = 4.702779672 x2 = 4.702779672 Simplifying x2 = 4.702779672 Take the square root of each side: x = {-2.168589328, 2.168589328}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-6.196623094, 6.196623094, -2.168589328, 2.168589328}Solution
x = {-6.196623094, 6.196623094, -2.168589328, 2.168589328}
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