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Simplifying 4x(27x + 6) = 35 Reorder the terms: 4x(6 + 27x) = 35 (6 * 4x + 27x * 4x) = 35 (24x + 108x2) = 35 Solving 24x + 108x2 = 35 Solving for variable 'x'. Reorder the terms: -35 + 24x + 108x2 = 35 + -35 Combine like terms: 35 + -35 = 0 -35 + 24x + 108x2 = 0 Begin completing the square. Divide all terms by 108 the coefficient of the squared term: Divide each side by '108'. -0.3240740741 + 0.2222222222x + x2 = 0 Move the constant term to the right: Add '0.3240740741' to each side of the equation. -0.3240740741 + 0.2222222222x + 0.3240740741 + x2 = 0 + 0.3240740741 Reorder the terms: -0.3240740741 + 0.3240740741 + 0.2222222222x + x2 = 0 + 0.3240740741 Combine like terms: -0.3240740741 + 0.3240740741 = 0.0000000000 0.0000000000 + 0.2222222222x + x2 = 0 + 0.3240740741 0.2222222222x + x2 = 0 + 0.3240740741 Combine like terms: 0 + 0.3240740741 = 0.3240740741 0.2222222222x + x2 = 0.3240740741 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 = 0.3240740741 + 0.01234567901 Reorder the terms: 0.01234567901 + 0.2222222222x + x2 = 0.3240740741 + 0.01234567901 Combine like terms: 0.3240740741 + 0.01234567901 = 0.33641975311 0.01234567901 + 0.2222222222x + x2 = 0.33641975311 Factor a perfect square on the left side: (x + 0.1111111111)(x + 0.1111111111) = 0.33641975311 Calculate the square root of the right side: 0.580017028 Break this problem into two subproblems by setting (x + 0.1111111111) equal to 0.580017028 and -0.580017028.Subproblem 1
x + 0.1111111111 = 0.580017028 Simplifying x + 0.1111111111 = 0.580017028 Reorder the terms: 0.1111111111 + x = 0.580017028 Solving 0.1111111111 + x = 0.580017028 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 = 0.580017028 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + x = 0.580017028 + -0.1111111111 x = 0.580017028 + -0.1111111111 Combine like terms: 0.580017028 + -0.1111111111 = 0.4689059169 x = 0.4689059169 Simplifying x = 0.4689059169Subproblem 2
x + 0.1111111111 = -0.580017028 Simplifying x + 0.1111111111 = -0.580017028 Reorder the terms: 0.1111111111 + x = -0.580017028 Solving 0.1111111111 + x = -0.580017028 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 = -0.580017028 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + x = -0.580017028 + -0.1111111111 x = -0.580017028 + -0.1111111111 Combine like terms: -0.580017028 + -0.1111111111 = -0.6911281391 x = -0.6911281391 Simplifying x = -0.6911281391Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.4689059169, -0.6911281391}
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