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Simplifying 0 = 3x2 + 6x + 1 Reorder the terms: 0 = 1 + 6x + 3x2 Solving 0 = 1 + 6x + 3x2 Solving for variable 'x'. Combine like terms: 0 + -1 = -1 -1 + -6x + -3x2 = 1 + 6x + 3x2 + -1 + -6x + -3x2 Reorder the terms: -1 + -6x + -3x2 = 1 + -1 + 6x + -6x + 3x2 + -3x2 Combine like terms: 1 + -1 = 0 -1 + -6x + -3x2 = 0 + 6x + -6x + 3x2 + -3x2 -1 + -6x + -3x2 = 6x + -6x + 3x2 + -3x2 Combine like terms: 6x + -6x = 0 -1 + -6x + -3x2 = 0 + 3x2 + -3x2 -1 + -6x + -3x2 = 3x2 + -3x2 Combine like terms: 3x2 + -3x2 = 0 -1 + -6x + -3x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 6x + 3x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 6x + 3x2)' equal to zero and attempt to solve: Simplifying 1 + 6x + 3x2 = 0 Solving 1 + 6x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.3333333333 + 2x + x2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + 2x + -0.3333333333 + x2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + 2x + x2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + 2x + x2 = 0 + -0.3333333333 2x + x2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 2x + x2 = -0.3333333333 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = -0.3333333333 + 1 Reorder the terms: 1 + 2x + x2 = -0.3333333333 + 1 Combine like terms: -0.3333333333 + 1 = 0.6666666667 1 + 2x + x2 = 0.6666666667 Factor a perfect square on the left side: (x + 1)(x + 1) = 0.6666666667 Calculate the square root of the right side: 0.816496581 Break this problem into two subproblems by setting (x + 1) equal to 0.816496581 and -0.816496581.Subproblem 1
x + 1 = 0.816496581 Simplifying x + 1 = 0.816496581 Reorder the terms: 1 + x = 0.816496581 Solving 1 + x = 0.816496581 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 0.816496581 + -1 Combine like terms: 1 + -1 = 0 0 + x = 0.816496581 + -1 x = 0.816496581 + -1 Combine like terms: 0.816496581 + -1 = -0.183503419 x = -0.183503419 Simplifying x = -0.183503419Subproblem 2
x + 1 = -0.816496581 Simplifying x + 1 = -0.816496581 Reorder the terms: 1 + x = -0.816496581 Solving 1 + x = -0.816496581 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -0.816496581 + -1 Combine like terms: 1 + -1 = 0 0 + x = -0.816496581 + -1 x = -0.816496581 + -1 Combine like terms: -0.816496581 + -1 = -1.816496581 x = -1.816496581 Simplifying x = -1.816496581Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.183503419, -1.816496581}Solution
x = {-0.183503419, -1.816496581}
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