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Simplifying (3x * x) + 6x + 2 = 0 Multiply x * x (3x2) + 6x + 2 = 0 Reorder the terms: 2 + 6x + (3x2) = 0 Solving 2 + 6x + (3x2) = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.6666666667 + 2x + x2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + 2x + -0.6666666667 + x2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + 2x + x2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + 2x + x2 = 0 + -0.6666666667 2x + x2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 2x + x2 = -0.6666666667 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 = -0.6666666667 + 1 Reorder the terms: 1 + 2x + x2 = -0.6666666667 + 1 Combine like terms: -0.6666666667 + 1 = 0.3333333333 1 + 2x + x2 = 0.3333333333 Factor a perfect square on the left side: ((x) + 1)((x) + 1) = 0.3333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting ((x) + 1) equal to 0.577350269 and -0.577350269.Subproblem 1
(x) + 1 = 0.577350269 Simplifying (x) + 1 = 0.577350269 x + 1 = 0.577350269 Reorder the terms: 1 + x = 0.577350269 Solving 1 + x = 0.577350269 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 = 0.577350269 + -1 Combine like terms: 1 + -1 = 0 0 + x = 0.577350269 + -1 x = 0.577350269 + -1 Combine like terms: 0.577350269 + -1 = -0.422649731 x = -0.422649731 Simplifying x = -0.422649731Subproblem 2
(x) + 1 = -0.577350269 Simplifying (x) + 1 = -0.577350269 x + 1 = -0.577350269 Reorder the terms: 1 + x = -0.577350269 Solving 1 + x = -0.577350269 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 = -0.577350269 + -1 Combine like terms: 1 + -1 = 0 0 + x = -0.577350269 + -1 x = -0.577350269 + -1 Combine like terms: -0.577350269 + -1 = -1.577350269 x = -1.577350269 Simplifying x = -1.577350269Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.422649731, -1.577350269}
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