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Simplifying 12x3 + -10x2 + -16x = 0 Reorder the terms: -16x + -10x2 + 12x3 = 0 Solving -16x + -10x2 + 12x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2x'. 2x(-8 + -5x + 6x2) = 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 '(-8 + -5x + 6x2)' equal to zero and attempt to solve: Simplifying -8 + -5x + 6x2 = 0 Solving -8 + -5x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.333333333 + -0.8333333333x + x2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -0.8333333333x + 1.333333333 + x2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -0.8333333333x + x2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -0.8333333333x + x2 = 0 + 1.333333333 -0.8333333333x + x2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -0.8333333333x + x2 = 1.333333333 The x term is -0.8333333333x. Take half its coefficient (-0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. -0.8333333333x + 0.1736111111 + x2 = 1.333333333 + 0.1736111111 Reorder the terms: 0.1736111111 + -0.8333333333x + x2 = 1.333333333 + 0.1736111111 Combine like terms: 1.333333333 + 0.1736111111 = 1.5069444441 0.1736111111 + -0.8333333333x + x2 = 1.5069444441 Factor a perfect square on the left side: (x + -0.4166666667)(x + -0.4166666667) = 1.5069444441 Calculate the square root of the right side: 1.227576655 Break this problem into two subproblems by setting (x + -0.4166666667) equal to 1.227576655 and -1.227576655.Subproblem 1
x + -0.4166666667 = 1.227576655 Simplifying x + -0.4166666667 = 1.227576655 Reorder the terms: -0.4166666667 + x = 1.227576655 Solving -0.4166666667 + x = 1.227576655 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = 1.227576655 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = 1.227576655 + 0.4166666667 x = 1.227576655 + 0.4166666667 Combine like terms: 1.227576655 + 0.4166666667 = 1.6442433217 x = 1.6442433217 Simplifying x = 1.6442433217Subproblem 2
x + -0.4166666667 = -1.227576655 Simplifying x + -0.4166666667 = -1.227576655 Reorder the terms: -0.4166666667 + x = -1.227576655 Solving -0.4166666667 + x = -1.227576655 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = -1.227576655 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = -1.227576655 + 0.4166666667 x = -1.227576655 + 0.4166666667 Combine like terms: -1.227576655 + 0.4166666667 = -0.8109099883 x = -0.8109099883 Simplifying x = -0.8109099883Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.6442433217, -0.8109099883}Solution
x = {0, 1.6442433217, -0.8109099883}
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