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Simplifying 27x2 + 6x + -3 = 0 Reorder the terms: -3 + 6x + 27x2 = 0 Solving -3 + 6x + 27x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + 2x + 9x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + 2x + 9x2)' equal to zero and attempt to solve: Simplifying -1 + 2x + 9x2 = 0 Solving -1 + 2x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + 0.2222222222x + x2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + 0.2222222222x + 0.1111111111 + x2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + 0.2222222222x + x2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + 0.2222222222x + x2 = 0 + 0.1111111111 0.2222222222x + x2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 0.2222222222x + x2 = 0.1111111111 The x term is 0.2222222222x. Take half its coefficient (0.1111111111). Square it (0.01234567901) and add it to both sides. Add '0.01234567901' to each side of the equation. 0.2222222222x + 0.01234567901 + x2 = 0.1111111111 + 0.01234567901 Reorder the terms: 0.01234567901 + 0.2222222222x + x2 = 0.1111111111 + 0.01234567901 Combine like terms: 0.1111111111 + 0.01234567901 = 0.12345679011 0.01234567901 + 0.2222222222x + x2 = 0.12345679011 Factor a perfect square on the left side: (x + 0.1111111111)(x + 0.1111111111) = 0.12345679011 Calculate the square root of the right side: 0.351364184 Break this problem into two subproblems by setting (x + 0.1111111111) equal to 0.351364184 and -0.351364184.Subproblem 1
x + 0.1111111111 = 0.351364184 Simplifying x + 0.1111111111 = 0.351364184 Reorder the terms: 0.1111111111 + x = 0.351364184 Solving 0.1111111111 + x = 0.351364184 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + x = 0.351364184 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + x = 0.351364184 + -0.1111111111 x = 0.351364184 + -0.1111111111 Combine like terms: 0.351364184 + -0.1111111111 = 0.2402530729 x = 0.2402530729 Simplifying x = 0.2402530729Subproblem 2
x + 0.1111111111 = -0.351364184 Simplifying x + 0.1111111111 = -0.351364184 Reorder the terms: 0.1111111111 + x = -0.351364184 Solving 0.1111111111 + x = -0.351364184 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + x = -0.351364184 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + x = -0.351364184 + -0.1111111111 x = -0.351364184 + -0.1111111111 Combine like terms: -0.351364184 + -0.1111111111 = -0.4624752951 x = -0.4624752951 Simplifying x = -0.4624752951Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.2402530729, -0.4624752951}Solution
x = {0.2402530729, -0.4624752951}
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