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Simplifying 3x4 + -2x3 + -4x2 = 0 Reorder the terms: -4x2 + -2x3 + 3x4 = 0 Solving -4x2 + -2x3 + 3x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(-4 + -2x + 3x2) = 0Subproblem 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 '(-4 + -2x + 3x2)' equal to zero and attempt to solve: Simplifying -4 + -2x + 3x2 = 0 Solving -4 + -2x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.333333333 + -0.6666666667x + x2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -0.6666666667x + 1.333333333 + x2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -0.6666666667x + x2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -0.6666666667x + x2 = 0 + 1.333333333 -0.6666666667x + x2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -0.6666666667x + x2 = 1.333333333 The x term is -0.6666666667x. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667x + 0.1111111112 + x2 = 1.333333333 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667x + x2 = 1.333333333 + 0.1111111112 Combine like terms: 1.333333333 + 0.1111111112 = 1.4444444442 0.1111111112 + -0.6666666667x + x2 = 1.4444444442 Factor a perfect square on the left side: (x + -0.3333333334)(x + -0.3333333334) = 1.4444444442 Calculate the square root of the right side: 1.201850425 Break this problem into two subproblems by setting (x + -0.3333333334) equal to 1.201850425 and -1.201850425.Subproblem 1
x + -0.3333333334 = 1.201850425 Simplifying x + -0.3333333334 = 1.201850425 Reorder the terms: -0.3333333334 + x = 1.201850425 Solving -0.3333333334 + x = 1.201850425 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = 1.201850425 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 1.201850425 + 0.3333333334 x = 1.201850425 + 0.3333333334 Combine like terms: 1.201850425 + 0.3333333334 = 1.5351837584 x = 1.5351837584 Simplifying x = 1.5351837584Subproblem 2
x + -0.3333333334 = -1.201850425 Simplifying x + -0.3333333334 = -1.201850425 Reorder the terms: -0.3333333334 + x = -1.201850425 Solving -0.3333333334 + x = -1.201850425 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = -1.201850425 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -1.201850425 + 0.3333333334 x = -1.201850425 + 0.3333333334 Combine like terms: -1.201850425 + 0.3333333334 = -0.8685170916 x = -0.8685170916 Simplifying x = -0.8685170916Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5351837584, -0.8685170916}Solution
x = {0, 1.5351837584, -0.8685170916}
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