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Simplifying 4x3 + 12x2 + 7x = 0 Reorder the terms: 7x + 12x2 + 4x3 = 0 Solving 7x + 12x2 + 4x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x'. x(7 + 12x + 4x2) = 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 + 12x + 4x2)' equal to zero and attempt to solve: Simplifying 7 + 12x + 4x2 = 0 Solving 7 + 12x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.75 + 3x + x2 = 0 Move the constant term to the right: Add '-1.75' to each side of the equation. 1.75 + 3x + -1.75 + x2 = 0 + -1.75 Reorder the terms: 1.75 + -1.75 + 3x + x2 = 0 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + 3x + x2 = 0 + -1.75 3x + x2 = 0 + -1.75 Combine like terms: 0 + -1.75 = -1.75 3x + x2 = -1.75 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 = -1.75 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -1.75 + 2.25 Combine like terms: -1.75 + 2.25 = 0.5 2.25 + 3x + x2 = 0.5 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (x + 1.5) equal to 0.707106781 and -0.707106781.Subproblem 1
x + 1.5 = 0.707106781 Simplifying x + 1.5 = 0.707106781 Reorder the terms: 1.5 + x = 0.707106781 Solving 1.5 + x = 0.707106781 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 = 0.707106781 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 0.707106781 + -1.5 x = 0.707106781 + -1.5 Combine like terms: 0.707106781 + -1.5 = -0.792893219 x = -0.792893219 Simplifying x = -0.792893219Subproblem 2
x + 1.5 = -0.707106781 Simplifying x + 1.5 = -0.707106781 Reorder the terms: 1.5 + x = -0.707106781 Solving 1.5 + x = -0.707106781 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 = -0.707106781 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -0.707106781 + -1.5 x = -0.707106781 + -1.5 Combine like terms: -0.707106781 + -1.5 = -2.207106781 x = -2.207106781 Simplifying x = -2.207106781Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.792893219, -2.207106781}Solution
x = {0, -0.792893219, -2.207106781}
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