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Simplifying 12x3 + 20x2 + -36x4 = 0 Reorder the terms: 20x2 + 12x3 + -36x4 = 0 Solving 20x2 + 12x3 + -36x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4x2'. 4x2(5 + 3x + -9x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve: Simplifying x2 = 0 Solving x2 = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x2 = 0 Take the square root of each side: x = {0}Subproblem 2
Set the factor '(5 + 3x + -9x2)' equal to zero and attempt to solve: Simplifying 5 + 3x + -9x2 = 0 Solving 5 + 3x + -9x2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.5555555556 + -0.3333333333x + x2 = 0 Move the constant term to the right: Add '0.5555555556' to each side of the equation. -0.5555555556 + -0.3333333333x + 0.5555555556 + x2 = 0 + 0.5555555556 Reorder the terms: -0.5555555556 + 0.5555555556 + -0.3333333333x + x2 = 0 + 0.5555555556 Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000 0.0000000000 + -0.3333333333x + x2 = 0 + 0.5555555556 -0.3333333333x + x2 = 0 + 0.5555555556 Combine like terms: 0 + 0.5555555556 = 0.5555555556 -0.3333333333x + x2 = 0.5555555556 The x term is -0.3333333333x. Take half its coefficient (-0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. -0.3333333333x + 0.02777777779 + x2 = 0.5555555556 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333x + x2 = 0.5555555556 + 0.02777777779 Combine like terms: 0.5555555556 + 0.02777777779 = 0.58333333339 0.02777777779 + -0.3333333333x + x2 = 0.58333333339 Factor a perfect square on the left side: (x + -0.1666666667)(x + -0.1666666667) = 0.58333333339 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (x + -0.1666666667) equal to 0.763762616 and -0.763762616.Subproblem 1
x + -0.1666666667 = 0.763762616 Simplifying x + -0.1666666667 = 0.763762616 Reorder the terms: -0.1666666667 + x = 0.763762616 Solving -0.1666666667 + x = 0.763762616 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + x = 0.763762616 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + x = 0.763762616 + 0.1666666667 x = 0.763762616 + 0.1666666667 Combine like terms: 0.763762616 + 0.1666666667 = 0.9304292827 x = 0.9304292827 Simplifying x = 0.9304292827Subproblem 2
x + -0.1666666667 = -0.763762616 Simplifying x + -0.1666666667 = -0.763762616 Reorder the terms: -0.1666666667 + x = -0.763762616 Solving -0.1666666667 + x = -0.763762616 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + x = -0.763762616 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + x = -0.763762616 + 0.1666666667 x = -0.763762616 + 0.1666666667 Combine like terms: -0.763762616 + 0.1666666667 = -0.5970959493 x = -0.5970959493 Simplifying x = -0.5970959493Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.9304292827, -0.5970959493}Solution
x = {0, 0.9304292827, -0.5970959493}
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