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Simplifying 12x3 + 32x2 + 7x = 0 Reorder the terms: 7x + 32x2 + 12x3 = 0 Solving 7x + 32x2 + 12x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(7 + 32x + 12x2) = 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 '(7 + 32x + 12x2)' equal to zero and attempt to solve: Simplifying 7 + 32x + 12x2 = 0 Solving 7 + 32x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.5833333333 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '-0.5833333333' to each side of the equation. 0.5833333333 + 2.666666667x + -0.5833333333 + x2 = 0 + -0.5833333333 Reorder the terms: 0.5833333333 + -0.5833333333 + 2.666666667x + x2 = 0 + -0.5833333333 Combine like terms: 0.5833333333 + -0.5833333333 = 0.0000000000 0.0000000000 + 2.666666667x + x2 = 0 + -0.5833333333 2.666666667x + x2 = 0 + -0.5833333333 Combine like terms: 0 + -0.5833333333 = -0.5833333333 2.666666667x + x2 = -0.5833333333 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = -0.5833333333 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = -0.5833333333 + 1.777777780 Combine like terms: -0.5833333333 + 1.777777780 = 1.1944444467 1.777777780 + 2.666666667x + x2 = 1.1944444467 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 1.1944444467 Calculate the square root of the right side: 1.092906422 Break this problem into two subproblems by setting (x + 1.333333334) equal to 1.092906422 and -1.092906422.Subproblem 1
x + 1.333333334 = 1.092906422 Simplifying x + 1.333333334 = 1.092906422 Reorder the terms: 1.333333334 + x = 1.092906422 Solving 1.333333334 + x = 1.092906422 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 1.092906422 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 1.092906422 + -1.333333334 x = 1.092906422 + -1.333333334 Combine like terms: 1.092906422 + -1.333333334 = -0.240426912 x = -0.240426912 Simplifying x = -0.240426912Subproblem 2
x + 1.333333334 = -1.092906422 Simplifying x + 1.333333334 = -1.092906422 Reorder the terms: 1.333333334 + x = -1.092906422 Solving 1.333333334 + x = -1.092906422 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -1.092906422 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -1.092906422 + -1.333333334 x = -1.092906422 + -1.333333334 Combine like terms: -1.092906422 + -1.333333334 = -2.426239756 x = -2.426239756 Simplifying x = -2.426239756Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.240426912, -2.426239756}Solution
x = {0, -0.240426912, -2.426239756}
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