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Simplifying 4x3 + 12x2 + 6x = 0 Reorder the terms: 6x + 12x2 + 4x3 = 0 Solving 6x + 12x2 + 4x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2x'. 2x(3 + 6x + 2x2) = 0 Ignore the factor 2.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 '(3 + 6x + 2x2)' equal to zero and attempt to solve: Simplifying 3 + 6x + 2x2 = 0 Solving 3 + 6x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + 3x + x2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 3x + -1.5 + x2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 3x + x2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 3x + x2 = 0 + -1.5 3x + x2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 3x + x2 = -1.5 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = -1.5 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + 3x + x2 = 0.75 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 1.5 = 0.866025404 Simplifying x + 1.5 = 0.866025404 Reorder the terms: 1.5 + x = 0.866025404 Solving 1.5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 0.866025404 + -1.5 x = 0.866025404 + -1.5 Combine like terms: 0.866025404 + -1.5 = -0.633974596 x = -0.633974596 Simplifying x = -0.633974596Subproblem 2
x + 1.5 = -0.866025404 Simplifying x + 1.5 = -0.866025404 Reorder the terms: 1.5 + x = -0.866025404 Solving 1.5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -0.866025404 + -1.5 x = -0.866025404 + -1.5 Combine like terms: -0.866025404 + -1.5 = -2.366025404 x = -2.366025404 Simplifying x = -2.366025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.633974596, -2.366025404}Solution
x = {0, -0.633974596, -2.366025404}
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