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Simplifying 6x(x + 1) = (x + 3) * 2 Reorder the terms: 6x(1 + x) = (x + 3) * 2 (1 * 6x + x * 6x) = (x + 3) * 2 (6x + 6x2) = (x + 3) * 2 Reorder the terms: 6x + 6x2 = (3 + x) * 2 Reorder the terms for easier multiplication: 6x + 6x2 = 2(3 + x) 6x + 6x2 = (3 * 2 + x * 2) 6x + 6x2 = (6 + 2x) Solving 6x + 6x2 = 6 + 2x Solving for variable 'x'. Reorder the terms: -6 + 6x + -2x + 6x2 = 6 + 2x + -6 + -2x Combine like terms: 6x + -2x = 4x -6 + 4x + 6x2 = 6 + 2x + -6 + -2x Reorder the terms: -6 + 4x + 6x2 = 6 + -6 + 2x + -2x Combine like terms: 6 + -6 = 0 -6 + 4x + 6x2 = 0 + 2x + -2x -6 + 4x + 6x2 = 2x + -2x Combine like terms: 2x + -2x = 0 -6 + 4x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 2x + 3x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 2x + 3x2)' equal to zero and attempt to solve: Simplifying -3 + 2x + 3x2 = 0 Solving -3 + 2x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + 0.6666666667x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.6666666667x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + 0.6666666667x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.6666666667x + x2 = 0 + 1 0.6666666667x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.6666666667x + x2 = 1 The x term is 0.6666666667x. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667x + 0.1111111112 + x2 = 1 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667x + x2 = 1 + 0.1111111112 Combine like terms: 1 + 0.1111111112 = 1.1111111112 0.1111111112 + 0.6666666667x + x2 = 1.1111111112 Factor a perfect square on the left side: (x + 0.3333333334)(x + 0.3333333334) = 1.1111111112 Calculate the square root of the right side: 1.054092553 Break this problem into two subproblems by setting (x + 0.3333333334) equal to 1.054092553 and -1.054092553.Subproblem 1
x + 0.3333333334 = 1.054092553 Simplifying x + 0.3333333334 = 1.054092553 Reorder the terms: 0.3333333334 + x = 1.054092553 Solving 0.3333333334 + x = 1.054092553 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 1.054092553 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 1.054092553 + -0.3333333334 x = 1.054092553 + -0.3333333334 Combine like terms: 1.054092553 + -0.3333333334 = 0.7207592196 x = 0.7207592196 Simplifying x = 0.7207592196Subproblem 2
x + 0.3333333334 = -1.054092553 Simplifying x + 0.3333333334 = -1.054092553 Reorder the terms: 0.3333333334 + x = -1.054092553 Solving 0.3333333334 + x = -1.054092553 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -1.054092553 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -1.054092553 + -0.3333333334 x = -1.054092553 + -0.3333333334 Combine like terms: -1.054092553 + -0.3333333334 = -1.3874258864 x = -1.3874258864 Simplifying x = -1.3874258864Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7207592196, -1.3874258864}Solution
x = {0.7207592196, -1.3874258864}
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