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Simplifying (3x3 + -7x2 + 12) + -1(3x3 + 6x2 + 10x) = 0 Reorder the terms: (12 + -7x2 + 3x3) + -1(3x3 + 6x2 + 10x) = 0 Remove parenthesis around (12 + -7x2 + 3x3) 12 + -7x2 + 3x3 + -1(3x3 + 6x2 + 10x) = 0 Reorder the terms: 12 + -7x2 + 3x3 + -1(10x + 6x2 + 3x3) = 0 12 + -7x2 + 3x3 + (10x * -1 + 6x2 * -1 + 3x3 * -1) = 0 12 + -7x2 + 3x3 + (-10x + -6x2 + -3x3) = 0 Reorder the terms: 12 + -10x + -7x2 + -6x2 + 3x3 + -3x3 = 0 Combine like terms: -7x2 + -6x2 = -13x2 12 + -10x + -13x2 + 3x3 + -3x3 = 0 Combine like terms: 3x3 + -3x3 = 0 12 + -10x + -13x2 + 0 = 0 12 + -10x + -13x2 = 0 Solving 12 + -10x + -13x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -13 the coefficient of the squared term: Divide each side by '-13'. -0.9230769231 + 0.7692307692x + x2 = 0 Move the constant term to the right: Add '0.9230769231' to each side of the equation. -0.9230769231 + 0.7692307692x + 0.9230769231 + x2 = 0 + 0.9230769231 Reorder the terms: -0.9230769231 + 0.9230769231 + 0.7692307692x + x2 = 0 + 0.9230769231 Combine like terms: -0.9230769231 + 0.9230769231 = 0.0000000000 0.0000000000 + 0.7692307692x + x2 = 0 + 0.9230769231 0.7692307692x + x2 = 0 + 0.9230769231 Combine like terms: 0 + 0.9230769231 = 0.9230769231 0.7692307692x + x2 = 0.9230769231 The x term is 0.7692307692x. Take half its coefficient (0.3846153846). Square it (0.1479289941) and add it to both sides. Add '0.1479289941' to each side of the equation. 0.7692307692x + 0.1479289941 + x2 = 0.9230769231 + 0.1479289941 Reorder the terms: 0.1479289941 + 0.7692307692x + x2 = 0.9230769231 + 0.1479289941 Combine like terms: 0.9230769231 + 0.1479289941 = 1.0710059172 0.1479289941 + 0.7692307692x + x2 = 1.0710059172 Factor a perfect square on the left side: (x + 0.3846153846)(x + 0.3846153846) = 1.0710059172 Calculate the square root of the right side: 1.034894157 Break this problem into two subproblems by setting (x + 0.3846153846) equal to 1.034894157 and -1.034894157.Subproblem 1
x + 0.3846153846 = 1.034894157 Simplifying x + 0.3846153846 = 1.034894157 Reorder the terms: 0.3846153846 + x = 1.034894157 Solving 0.3846153846 + x = 1.034894157 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3846153846' to each side of the equation. 0.3846153846 + -0.3846153846 + x = 1.034894157 + -0.3846153846 Combine like terms: 0.3846153846 + -0.3846153846 = 0.0000000000 0.0000000000 + x = 1.034894157 + -0.3846153846 x = 1.034894157 + -0.3846153846 Combine like terms: 1.034894157 + -0.3846153846 = 0.6502787724 x = 0.6502787724 Simplifying x = 0.6502787724Subproblem 2
x + 0.3846153846 = -1.034894157 Simplifying x + 0.3846153846 = -1.034894157 Reorder the terms: 0.3846153846 + x = -1.034894157 Solving 0.3846153846 + x = -1.034894157 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3846153846' to each side of the equation. 0.3846153846 + -0.3846153846 + x = -1.034894157 + -0.3846153846 Combine like terms: 0.3846153846 + -0.3846153846 = 0.0000000000 0.0000000000 + x = -1.034894157 + -0.3846153846 x = -1.034894157 + -0.3846153846 Combine like terms: -1.034894157 + -0.3846153846 = -1.4195095416 x = -1.4195095416 Simplifying x = -1.4195095416Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.6502787724, -1.4195095416}
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