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Simplifying (3x + -1)(4x + -2)(x) = x Reorder the terms: (-1 + 3x)(4x + -2)(x) = x Reorder the terms: (-1 + 3x)(-2 + 4x)(x) = x Reorder the terms for easier multiplication: x(-1 + 3x)(-2 + 4x) = x Multiply (-1 + 3x) * (-2 + 4x) x(-1(-2 + 4x) + 3x * (-2 + 4x)) = x x((-2 * -1 + 4x * -1) + 3x * (-2 + 4x)) = x x((2 + -4x) + 3x * (-2 + 4x)) = x x(2 + -4x + (-2 * 3x + 4x * 3x)) = x x(2 + -4x + (-6x + 12x2)) = x Combine like terms: -4x + -6x = -10x x(2 + -10x + 12x2) = x (2 * x + -10x * x + 12x2 * x) = x (2x + -10x2 + 12x3) = x Solving 2x + -10x2 + 12x3 = x Solving for variable 'x'. Reorder the terms: 2x + -1x + -10x2 + 12x3 = x + -1x Combine like terms: 2x + -1x = 1x 1x + -10x2 + 12x3 = x + -1x Combine like terms: x + -1x = 0 1x + -10x2 + 12x3 = 0 Factor out the Greatest Common Factor (GCF), 'x'. x(1 + -10x + 12x2) = 0Subproblem 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 '(1 + -10x + 12x2)' equal to zero and attempt to solve: Simplifying 1 + -10x + 12x2 = 0 Solving 1 + -10x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.08333333333 + -0.8333333333x + x2 = 0 Move the constant term to the right: Add '-0.08333333333' to each side of the equation. 0.08333333333 + -0.8333333333x + -0.08333333333 + x2 = 0 + -0.08333333333 Reorder the terms: 0.08333333333 + -0.08333333333 + -0.8333333333x + x2 = 0 + -0.08333333333 Combine like terms: 0.08333333333 + -0.08333333333 = 0.00000000000 0.00000000000 + -0.8333333333x + x2 = 0 + -0.08333333333 -0.8333333333x + x2 = 0 + -0.08333333333 Combine like terms: 0 + -0.08333333333 = -0.08333333333 -0.8333333333x + x2 = -0.08333333333 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 = -0.08333333333 + 0.1736111111 Reorder the terms: 0.1736111111 + -0.8333333333x + x2 = -0.08333333333 + 0.1736111111 Combine like terms: -0.08333333333 + 0.1736111111 = 0.09027777777 0.1736111111 + -0.8333333333x + x2 = 0.09027777777 Factor a perfect square on the left side: (x + -0.4166666667)(x + -0.4166666667) = 0.09027777777 Calculate the square root of the right side: 0.300462606 Break this problem into two subproblems by setting (x + -0.4166666667) equal to 0.300462606 and -0.300462606.Subproblem 1
x + -0.4166666667 = 0.300462606 Simplifying x + -0.4166666667 = 0.300462606 Reorder the terms: -0.4166666667 + x = 0.300462606 Solving -0.4166666667 + x = 0.300462606 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 = 0.300462606 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = 0.300462606 + 0.4166666667 x = 0.300462606 + 0.4166666667 Combine like terms: 0.300462606 + 0.4166666667 = 0.7171292727 x = 0.7171292727 Simplifying x = 0.7171292727Subproblem 2
x + -0.4166666667 = -0.300462606 Simplifying x + -0.4166666667 = -0.300462606 Reorder the terms: -0.4166666667 + x = -0.300462606 Solving -0.4166666667 + x = -0.300462606 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 = -0.300462606 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = -0.300462606 + 0.4166666667 x = -0.300462606 + 0.4166666667 Combine like terms: -0.300462606 + 0.4166666667 = 0.1162040607 x = 0.1162040607 Simplifying x = 0.1162040607Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7171292727, 0.1162040607}Solution
x = {0, 0.7171292727, 0.1162040607}
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