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Simplifying 12x3 + 24x2 + 6x = 0 Reorder the terms: 6x + 24x2 + 12x3 = 0 Solving 6x + 24x2 + 12x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '6x'. 6x(1 + 4x + 2x2) = 0 Ignore the factor 6.Subproblem 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 '(1 + 4x + 2x2)' equal to zero and attempt to solve: Simplifying 1 + 4x + 2x2 = 0 Solving 1 + 4x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + 2x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 2x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 2x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 2x + x2 = 0 + -0.5 2x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 2x + x2 = -0.5 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = -0.5 + 1 Reorder the terms: 1 + 2x + x2 = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 1 + 2x + x2 = 0.5 Factor a perfect square on the left side: (x + 1)(x + 1) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (x + 1) equal to 0.707106781 and -0.707106781.Subproblem 1
x + 1 = 0.707106781 Simplifying x + 1 = 0.707106781 Reorder the terms: 1 + x = 0.707106781 Solving 1 + x = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 0.707106781 + -1 Combine like terms: 1 + -1 = 0 0 + x = 0.707106781 + -1 x = 0.707106781 + -1 Combine like terms: 0.707106781 + -1 = -0.292893219 x = -0.292893219 Simplifying x = -0.292893219Subproblem 2
x + 1 = -0.707106781 Simplifying x + 1 = -0.707106781 Reorder the terms: 1 + x = -0.707106781 Solving 1 + x = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -0.707106781 + -1 Combine like terms: 1 + -1 = 0 0 + x = -0.707106781 + -1 x = -0.707106781 + -1 Combine like terms: -0.707106781 + -1 = -1.707106781 x = -1.707106781 Simplifying x = -1.707106781Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.292893219, -1.707106781}Solution
x = {0, -0.292893219, -1.707106781}
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