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Simplifying 3x4 + -5x3 + -1x2 = 0 Reorder the terms: -1x2 + -5x3 + 3x4 = 0 Solving -1x2 + -5x3 + 3x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(-1 + -5x + 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 '(-1 + -5x + 3x2)' equal to zero and attempt to solve: Simplifying -1 + -5x + 3x2 = 0 Solving -1 + -5x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + -1.666666667x + x2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -1.666666667x + 0.3333333333 + x2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -1.666666667x + x2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -1.666666667x + x2 = 0 + 0.3333333333 -1.666666667x + x2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -1.666666667x + x2 = 0.3333333333 The x term is -1.666666667x. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667x + 0.6944444447 + x2 = 0.3333333333 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667x + x2 = 0.3333333333 + 0.6944444447 Combine like terms: 0.3333333333 + 0.6944444447 = 1.027777778 0.6944444447 + -1.666666667x + x2 = 1.027777778 Factor a perfect square on the left side: (x + -0.8333333335)(x + -0.8333333335) = 1.027777778 Calculate the square root of the right side: 1.013793755 Break this problem into two subproblems by setting (x + -0.8333333335) equal to 1.013793755 and -1.013793755.Subproblem 1
x + -0.8333333335 = 1.013793755 Simplifying x + -0.8333333335 = 1.013793755 Reorder the terms: -0.8333333335 + x = 1.013793755 Solving -0.8333333335 + x = 1.013793755 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 1.013793755 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 1.013793755 + 0.8333333335 x = 1.013793755 + 0.8333333335 Combine like terms: 1.013793755 + 0.8333333335 = 1.8471270885 x = 1.8471270885 Simplifying x = 1.8471270885Subproblem 2
x + -0.8333333335 = -1.013793755 Simplifying x + -0.8333333335 = -1.013793755 Reorder the terms: -0.8333333335 + x = -1.013793755 Solving -0.8333333335 + x = -1.013793755 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -1.013793755 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -1.013793755 + 0.8333333335 x = -1.013793755 + 0.8333333335 Combine like terms: -1.013793755 + 0.8333333335 = -0.1804604215 x = -0.1804604215 Simplifying x = -0.1804604215Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.8471270885, -0.1804604215}Solution
x = {0, 1.8471270885, -0.1804604215}
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