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Simplifying 54x2 + 18x + -10 = 0 Reorder the terms: -10 + 18x + 54x2 = 0 Solving -10 + 18x + 54x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + 9x + 27x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + 9x + 27x2)' equal to zero and attempt to solve: Simplifying -5 + 9x + 27x2 = 0 Solving -5 + 9x + 27x2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.1851851852 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '0.1851851852' to each side of the equation. -0.1851851852 + 0.3333333333x + 0.1851851852 + x2 = 0 + 0.1851851852 Reorder the terms: -0.1851851852 + 0.1851851852 + 0.3333333333x + x2 = 0 + 0.1851851852 Combine like terms: -0.1851851852 + 0.1851851852 = 0.0000000000 0.0000000000 + 0.3333333333x + x2 = 0 + 0.1851851852 0.3333333333x + x2 = 0 + 0.1851851852 Combine like terms: 0 + 0.1851851852 = 0.1851851852 0.3333333333x + x2 = 0.1851851852 The x term is 0.3333333333x. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333x + 0.02777777779 + x2 = 0.1851851852 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333x + x2 = 0.1851851852 + 0.02777777779 Combine like terms: 0.1851851852 + 0.02777777779 = 0.21296296299 0.02777777779 + 0.3333333333x + x2 = 0.21296296299 Factor a perfect square on the left side: (x + 0.1666666667)(x + 0.1666666667) = 0.21296296299 Calculate the square root of the right side: 0.461479104 Break this problem into two subproblems by setting (x + 0.1666666667) equal to 0.461479104 and -0.461479104.Subproblem 1
x + 0.1666666667 = 0.461479104 Simplifying x + 0.1666666667 = 0.461479104 Reorder the terms: 0.1666666667 + x = 0.461479104 Solving 0.1666666667 + x = 0.461479104 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = 0.461479104 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 0.461479104 + -0.1666666667 x = 0.461479104 + -0.1666666667 Combine like terms: 0.461479104 + -0.1666666667 = 0.2948124373 x = 0.2948124373 Simplifying x = 0.2948124373Subproblem 2
x + 0.1666666667 = -0.461479104 Simplifying x + 0.1666666667 = -0.461479104 Reorder the terms: 0.1666666667 + x = -0.461479104 Solving 0.1666666667 + x = -0.461479104 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = -0.461479104 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -0.461479104 + -0.1666666667 x = -0.461479104 + -0.1666666667 Combine like terms: -0.461479104 + -0.1666666667 = -0.6281457707 x = -0.6281457707 Simplifying x = -0.6281457707Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.2948124373, -0.6281457707}Solution
x = {0.2948124373, -0.6281457707}
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