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Simplifying 12x3 + 10x2 + -6x = 0 Reorder the terms: -6x + 10x2 + 12x3 = 0 Solving -6x + 10x2 + 12x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2x'. 2x(-3 + 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 '(-3 + 5x + 6x2)' equal to zero and attempt to solve: Simplifying -3 + 5x + 6x2 = 0 Solving -3 + 5x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.5 + 0.8333333333x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 0.8333333333x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 0.8333333333x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 0.8333333333x + x2 = 0 + 0.5 0.8333333333x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 0.8333333333x + x2 = 0.5 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 = 0.5 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333x + x2 = 0.5 + 0.1736111111 Combine like terms: 0.5 + 0.1736111111 = 0.6736111111 0.1736111111 + 0.8333333333x + x2 = 0.6736111111 Factor a perfect square on the left side: (x + 0.4166666667)(x + 0.4166666667) = 0.6736111111 Calculate the square root of the right side: 0.82073815 Break this problem into two subproblems by setting (x + 0.4166666667) equal to 0.82073815 and -0.82073815.Subproblem 1
x + 0.4166666667 = 0.82073815 Simplifying x + 0.4166666667 = 0.82073815 Reorder the terms: 0.4166666667 + x = 0.82073815 Solving 0.4166666667 + x = 0.82073815 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 = 0.82073815 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 0.82073815 + -0.4166666667 x = 0.82073815 + -0.4166666667 Combine like terms: 0.82073815 + -0.4166666667 = 0.4040714833 x = 0.4040714833 Simplifying x = 0.4040714833Subproblem 2
x + 0.4166666667 = -0.82073815 Simplifying x + 0.4166666667 = -0.82073815 Reorder the terms: 0.4166666667 + x = -0.82073815 Solving 0.4166666667 + x = -0.82073815 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 = -0.82073815 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -0.82073815 + -0.4166666667 x = -0.82073815 + -0.4166666667 Combine like terms: -0.82073815 + -0.4166666667 = -1.2374048167 x = -1.2374048167 Simplifying x = -1.2374048167Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.4040714833, -1.2374048167}Solution
x = {0, 0.4040714833, -1.2374048167}
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