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Simplifying 2(x + 3) * 6x + -4x + 6 = 0 Reorder the terms: 2(3 + x) * 6x + -4x + 6 = 0 Reorder the terms for easier multiplication: 2 * 6x(3 + x) + -4x + 6 = 0 Multiply 2 * 6 12x(3 + x) + -4x + 6 = 0 (3 * 12x + x * 12x) + -4x + 6 = 0 (36x + 12x2) + -4x + 6 = 0 Reorder the terms: 6 + 36x + -4x + 12x2 = 0 Combine like terms: 36x + -4x = 32x 6 + 32x + 12x2 = 0 Solving 6 + 32x + 12x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(3 + 16x + 6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(3 + 16x + 6x2)' equal to zero and attempt to solve: Simplifying 3 + 16x + 6x2 = 0 Solving 3 + 16x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.5 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 2.666666667x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 2.666666667x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 2.666666667x + x2 = 0 + -0.5 2.666666667x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 2.666666667x + x2 = -0.5 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = -0.5 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = -0.5 + 1.777777780 Combine like terms: -0.5 + 1.777777780 = 1.27777778 1.777777780 + 2.666666667x + x2 = 1.27777778 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 1.27777778 Calculate the square root of the right side: 1.130388332 Break this problem into two subproblems by setting (x + 1.333333334) equal to 1.130388332 and -1.130388332.Subproblem 1
x + 1.333333334 = 1.130388332 Simplifying x + 1.333333334 = 1.130388332 Reorder the terms: 1.333333334 + x = 1.130388332 Solving 1.333333334 + x = 1.130388332 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 1.130388332 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 1.130388332 + -1.333333334 x = 1.130388332 + -1.333333334 Combine like terms: 1.130388332 + -1.333333334 = -0.202945002 x = -0.202945002 Simplifying x = -0.202945002Subproblem 2
x + 1.333333334 = -1.130388332 Simplifying x + 1.333333334 = -1.130388332 Reorder the terms: 1.333333334 + x = -1.130388332 Solving 1.333333334 + x = -1.130388332 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -1.130388332 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -1.130388332 + -1.333333334 x = -1.130388332 + -1.333333334 Combine like terms: -1.130388332 + -1.333333334 = -2.463721666 x = -2.463721666 Simplifying x = -2.463721666Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.202945002, -2.463721666}Solution
x = {-0.202945002, -2.463721666}
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