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Simplifying 6x3 + 27x2 + 4x = 0 Reorder the terms: 4x + 27x2 + 6x3 = 0 Solving 4x + 27x2 + 6x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(4 + 27x + 6x2) = 0Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(4 + 27x + 6x2)' equal to zero and attempt to solve: Simplifying 4 + 27x + 6x2 = 0 Solving 4 + 27x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.6666666667 + 4.5x + x2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + 4.5x + -0.6666666667 + x2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + 4.5x + x2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + 4.5x + x2 = 0 + -0.6666666667 4.5x + x2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 4.5x + x2 = -0.6666666667 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = -0.6666666667 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = -0.6666666667 + 5.0625 Combine like terms: -0.6666666667 + 5.0625 = 4.3958333333 5.0625 + 4.5x + x2 = 4.3958333333 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 4.3958333333 Calculate the square root of the right side: 2.096624271 Break this problem into two subproblems by setting (x + 2.25) equal to 2.096624271 and -2.096624271.Subproblem 1
x + 2.25 = 2.096624271 Simplifying x + 2.25 = 2.096624271 Reorder the terms: 2.25 + x = 2.096624271 Solving 2.25 + x = 2.096624271 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 2.096624271 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 2.096624271 + -2.25 x = 2.096624271 + -2.25 Combine like terms: 2.096624271 + -2.25 = -0.153375729 x = -0.153375729 Simplifying x = -0.153375729Subproblem 2
x + 2.25 = -2.096624271 Simplifying x + 2.25 = -2.096624271 Reorder the terms: 2.25 + x = -2.096624271 Solving 2.25 + x = -2.096624271 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -2.096624271 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -2.096624271 + -2.25 x = -2.096624271 + -2.25 Combine like terms: -2.096624271 + -2.25 = -4.346624271 x = -4.346624271 Simplifying x = -4.346624271Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.153375729, -4.346624271}Solution
x = {0, -0.153375729, -4.346624271}
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