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Simplifying 6x4 + -10x3 + 3x2 = 0 Reorder the terms: 3x2 + -10x3 + 6x4 = 0 Solving 3x2 + -10x3 + 6x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(3 + -10x + 6x2) = 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 '(3 + -10x + 6x2)' equal to zero and attempt to solve: Simplifying 3 + -10x + 6x2 = 0 Solving 3 + -10x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.5 + -1.666666667x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -1.666666667x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -1.666666667x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -1.666666667x + x2 = 0 + -0.5 -1.666666667x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -1.666666667x + x2 = -0.5 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 = -0.5 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667x + x2 = -0.5 + 0.6944444447 Combine like terms: -0.5 + 0.6944444447 = 0.1944444447 0.6944444447 + -1.666666667x + x2 = 0.1944444447 Factor a perfect square on the left side: (x + -0.8333333335)(x + -0.8333333335) = 0.1944444447 Calculate the square root of the right side: 0.440958552 Break this problem into two subproblems by setting (x + -0.8333333335) equal to 0.440958552 and -0.440958552.Subproblem 1
x + -0.8333333335 = 0.440958552 Simplifying x + -0.8333333335 = 0.440958552 Reorder the terms: -0.8333333335 + x = 0.440958552 Solving -0.8333333335 + x = 0.440958552 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 = 0.440958552 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 0.440958552 + 0.8333333335 x = 0.440958552 + 0.8333333335 Combine like terms: 0.440958552 + 0.8333333335 = 1.2742918855 x = 1.2742918855 Simplifying x = 1.2742918855Subproblem 2
x + -0.8333333335 = -0.440958552 Simplifying x + -0.8333333335 = -0.440958552 Reorder the terms: -0.8333333335 + x = -0.440958552 Solving -0.8333333335 + x = -0.440958552 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 = -0.440958552 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -0.440958552 + 0.8333333335 x = -0.440958552 + 0.8333333335 Combine like terms: -0.440958552 + 0.8333333335 = 0.3923747815 x = 0.3923747815 Simplifying x = 0.3923747815Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.2742918855, 0.3923747815}Solution
x = {0, 1.2742918855, 0.3923747815}
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