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Simplifying 18x3 + -1(x + -2)(3x + -2)(6x + -2) = 112 Reorder the terms: 18x3 + -1(-2 + x)(3x + -2)(6x + -2) = 112 Reorder the terms: 18x3 + -1(-2 + x)(-2 + 3x)(6x + -2) = 112 Reorder the terms: 18x3 + -1(-2 + x)(-2 + 3x)(-2 + 6x) = 112 Multiply (-2 + x) * (-2 + 3x) 18x3 + -1(-2(-2 + 3x) + x(-2 + 3x))(-2 + 6x) = 112 18x3 + -1((-2 * -2 + 3x * -2) + x(-2 + 3x))(-2 + 6x) = 112 18x3 + -1((4 + -6x) + x(-2 + 3x))(-2 + 6x) = 112 18x3 + -1(4 + -6x + (-2 * x + 3x * x))(-2 + 6x) = 112 18x3 + -1(4 + -6x + (-2x + 3x2))(-2 + 6x) = 112 Combine like terms: -6x + -2x = -8x 18x3 + -1(4 + -8x + 3x2)(-2 + 6x) = 112 Multiply (4 + -8x + 3x2) * (-2 + 6x) 18x3 + -1(4(-2 + 6x) + -8x * (-2 + 6x) + 3x2 * (-2 + 6x)) = 112 18x3 + -1((-2 * 4 + 6x * 4) + -8x * (-2 + 6x) + 3x2 * (-2 + 6x)) = 112 18x3 + -1((-8 + 24x) + -8x * (-2 + 6x) + 3x2 * (-2 + 6x)) = 112 18x3 + -1(-8 + 24x + (-2 * -8x + 6x * -8x) + 3x2 * (-2 + 6x)) = 112 18x3 + -1(-8 + 24x + (16x + -48x2) + 3x2 * (-2 + 6x)) = 112 18x3 + -1(-8 + 24x + 16x + -48x2 + (-2 * 3x2 + 6x * 3x2)) = 112 18x3 + -1(-8 + 24x + 16x + -48x2 + (-6x2 + 18x3)) = 112 Combine like terms: 24x + 16x = 40x 18x3 + -1(-8 + 40x + -48x2 + -6x2 + 18x3) = 112 Combine like terms: -48x2 + -6x2 = -54x2 18x3 + -1(-8 + 40x + -54x2 + 18x3) = 112 18x3 + (-8 * -1 + 40x * -1 + -54x2 * -1 + 18x3 * -1) = 112 18x3 + (8 + -40x + 54x2 + -18x3) = 112 Reorder the terms: 8 + -40x + 54x2 + 18x3 + -18x3 = 112 Combine like terms: 18x3 + -18x3 = 0 8 + -40x + 54x2 + 0 = 112 8 + -40x + 54x2 = 112 Solving 8 + -40x + 54x2 = 112 Solving for variable 'x'. Reorder the terms: 8 + -112 + -40x + 54x2 = 112 + -112 Combine like terms: 8 + -112 = -104 -104 + -40x + 54x2 = 112 + -112 Combine like terms: 112 + -112 = 0 -104 + -40x + 54x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-52 + -20x + 27x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-52 + -20x + 27x2)' equal to zero and attempt to solve: Simplifying -52 + -20x + 27x2 = 0 Solving -52 + -20x + 27x2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -1.925925926 + -0.7407407407x + x2 = 0 Move the constant term to the right: Add '1.925925926' to each side of the equation. -1.925925926 + -0.7407407407x + 1.925925926 + x2 = 0 + 1.925925926 Reorder the terms: -1.925925926 + 1.925925926 + -0.7407407407x + x2 = 0 + 1.925925926 Combine like terms: -1.925925926 + 1.925925926 = 0.000000000 0.000000000 + -0.7407407407x + x2 = 0 + 1.925925926 -0.7407407407x + x2 = 0 + 1.925925926 Combine like terms: 0 + 1.925925926 = 1.925925926 -0.7407407407x + x2 = 1.925925926 The x term is -0.7407407407x. Take half its coefficient (-0.3703703704). Square it (0.1371742113) and add it to both sides. Add '0.1371742113' to each side of the equation. -0.7407407407x + 0.1371742113 + x2 = 1.925925926 + 0.1371742113 Reorder the terms: 0.1371742113 + -0.7407407407x + x2 = 1.925925926 + 0.1371742113 Combine like terms: 1.925925926 + 0.1371742113 = 2.0631001373 0.1371742113 + -0.7407407407x + x2 = 2.0631001373 Factor a perfect square on the left side: (x + -0.3703703704)(x + -0.3703703704) = 2.0631001373 Calculate the square root of the right side: 1.436349587 Break this problem into two subproblems by setting (x + -0.3703703704) equal to 1.436349587 and -1.436349587.Subproblem 1
x + -0.3703703704 = 1.436349587 Simplifying x + -0.3703703704 = 1.436349587 Reorder the terms: -0.3703703704 + x = 1.436349587 Solving -0.3703703704 + x = 1.436349587 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3703703704' to each side of the equation. -0.3703703704 + 0.3703703704 + x = 1.436349587 + 0.3703703704 Combine like terms: -0.3703703704 + 0.3703703704 = 0.0000000000 0.0000000000 + x = 1.436349587 + 0.3703703704 x = 1.436349587 + 0.3703703704 Combine like terms: 1.436349587 + 0.3703703704 = 1.8067199574 x = 1.8067199574 Simplifying x = 1.8067199574Subproblem 2
x + -0.3703703704 = -1.436349587 Simplifying x + -0.3703703704 = -1.436349587 Reorder the terms: -0.3703703704 + x = -1.436349587 Solving -0.3703703704 + x = -1.436349587 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3703703704' to each side of the equation. -0.3703703704 + 0.3703703704 + x = -1.436349587 + 0.3703703704 Combine like terms: -0.3703703704 + 0.3703703704 = 0.0000000000 0.0000000000 + x = -1.436349587 + 0.3703703704 x = -1.436349587 + 0.3703703704 Combine like terms: -1.436349587 + 0.3703703704 = -1.0659792166 x = -1.0659792166 Simplifying x = -1.0659792166Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.8067199574, -1.0659792166}Solution
x = {1.8067199574, -1.0659792166}
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