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Simplifying 27x2 + -18x + -78 = 0 Reorder the terms: -78 + -18x + 27x2 = 0 Solving -78 + -18x + 27x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3'. 3(-26 + -6x + 9x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-26 + -6x + 9x2)' equal to zero and attempt to solve: Simplifying -26 + -6x + 9x2 = 0 Solving -26 + -6x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.888888889 + -0.6666666667x + x2 = 0 Move the constant term to the right: Add '2.888888889' to each side of the equation. -2.888888889 + -0.6666666667x + 2.888888889 + x2 = 0 + 2.888888889 Reorder the terms: -2.888888889 + 2.888888889 + -0.6666666667x + x2 = 0 + 2.888888889 Combine like terms: -2.888888889 + 2.888888889 = 0.000000000 0.000000000 + -0.6666666667x + x2 = 0 + 2.888888889 -0.6666666667x + x2 = 0 + 2.888888889 Combine like terms: 0 + 2.888888889 = 2.888888889 -0.6666666667x + x2 = 2.888888889 The x term is -0.6666666667x. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667x + 0.1111111112 + x2 = 2.888888889 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667x + x2 = 2.888888889 + 0.1111111112 Combine like terms: 2.888888889 + 0.1111111112 = 3.0000000002 0.1111111112 + -0.6666666667x + x2 = 3.0000000002 Factor a perfect square on the left side: (x + -0.3333333334)(x + -0.3333333334) = 3.0000000002 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (x + -0.3333333334) equal to 1.732050808 and -1.732050808.Subproblem 1
x + -0.3333333334 = 1.732050808 Simplifying x + -0.3333333334 = 1.732050808 Reorder the terms: -0.3333333334 + x = 1.732050808 Solving -0.3333333334 + x = 1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = 1.732050808 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 1.732050808 + 0.3333333334 x = 1.732050808 + 0.3333333334 Combine like terms: 1.732050808 + 0.3333333334 = 2.0653841414 x = 2.0653841414 Simplifying x = 2.0653841414Subproblem 2
x + -0.3333333334 = -1.732050808 Simplifying x + -0.3333333334 = -1.732050808 Reorder the terms: -0.3333333334 + x = -1.732050808 Solving -0.3333333334 + x = -1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = -1.732050808 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -1.732050808 + 0.3333333334 x = -1.732050808 + 0.3333333334 Combine like terms: -1.732050808 + 0.3333333334 = -1.3987174746 x = -1.3987174746 Simplifying x = -1.3987174746Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.0653841414, -1.3987174746}Solution
x = {2.0653841414, -1.3987174746}
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